[ { "id": "https://authors.library.caltech.edu/records/p5d1z-ttt30", "eprint_id": 99867, "eprint_status": "archive", "datestamp": "2023-08-19 18:32:11", "lastmod": "2023-10-18 18:53:07", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Fast generalized DFTs for all finite groups", "ispublished": "unpub", "full_text_status": "public", "keywords": "Discrete Fourier Transform, finite group, algorithm", "note": "\u00a9 2019 IEEE. \n\nSupported by NSF grant CCF-1815607 and a Simons Foundation Investigator grant.\n\n
Submitted - 1901.02536.pdf
", "abstract": "For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^(\u03c9/2+ \u03f5)) operations, for any \u03f5 > 0. Here, \u03c9 is the exponent of matrix multiplication.", "date": "2019-11", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Piscataway, NJ", "pagerange": "793-805", "id_number": "CaltechAUTHORS:20191115-133902016", "isbn": "978-1-7281-4952-3", "book_title": "2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191115-133902016", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-1815607" }, { "agency": "Simons Foundation" } ] }, "doi": "10.1109/FOCS.2019.00052", "primary_object": { "basename": "1901.02536.pdf", "url": "https://authors.library.caltech.edu/records/p5d1z-ttt30/files/1901.02536.pdf" }, "resource_type": "book_section", "pub_year": "2019", "author_list": "Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/s12d0-wtv85", "eprint_id": 85736, "eprint_status": "archive", "datestamp": "2023-08-19 07:11:37", "lastmod": "2023-10-18 18:43:58", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hsu-Chloe Ching-Yun", "name": { "family": "Hsu", "given": "Chloe Ching-Yun" }, "orcid": "0000-0002-7743-3168" }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "A fast generalized DFT for finite groups of Lie type", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2018 SIAM. \n\nSupported by NSF grant CCF-1423544 and a Simons Foundation Investigator grant. \n\nWe thank the SODA 2018 referees for their careful reading of this paper and many useful comments.\n\nPublished - p1047-hsu.pdf
Submitted - 1707.00349.pdf
", "abstract": "We give an arithmetic algorithm using O(|G|^(\u03c9/2+o(1))) operations to compute the generalized Discrete Fourier Transform (DFT) over group G for finite groups of Lie type, including the linear, orthogonal, and symplectic families and their variants, as well as all finite simple groups of Lie type. Here \u03c9 is the exponent of matrix multiplication, so the exponent \u03c9/2 is optimal if \u03c9 = 2. \n\nPreviously, \"exponent one\" algorithms were known for supersolvable groups and the symmetric and alternating groups. No exponent one algorithms were known (even under the assumption \u03c9 = 2) for families of linear groups of fixed dimension, and indeed the previous best-known algorithm for SL_2(F_q) had exponent 4/3 despite being the focus of significant effort. We unconditionally achieve exponent at most 1.19 for this group, and exponent one if \u03c9 = 2. \n\nWe also show that \u03c9 = 2 implies a \u221a2] exponent for general finite groups G, which beats the longstanding previous best upper bound (assuming \u03c9 = 2) of 3/2.", "date": "2018-01", "date_type": "published", "publisher": "Society for Industrial and Applied Mathematics", "place_of_pub": "Philadelphia, PA", "pagerange": "1047-1059", "id_number": "CaltechAUTHORS:20180410-153010180", "isbn": "978-1-6119-7503-1", "book_title": "Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180410-153010180", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-1423544" }, { "agency": "Simons Foundation" } ] }, "doi": "10.48550/arXiv.1707.00349", "primary_object": { "basename": "1707.00349.pdf", "url": "https://authors.library.caltech.edu/records/s12d0-wtv85/files/1707.00349.pdf" }, "related_objects": [ { "basename": "p1047-hsu.pdf", "url": "https://authors.library.caltech.edu/records/s12d0-wtv85/files/p1047-hsu.pdf" } ], "resource_type": "book_section", "pub_year": "2018", "author_list": "Hsu, Chloe Ching-Yun and Umans, Chris" }, { "id": "https://authors.library.caltech.edu/records/zf3cs-98987", "eprint_id": 99902, "eprint_status": "archive", "datestamp": "2023-08-19 05:28:19", "lastmod": "2023-10-18 18:54:36", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "FOCS 2017 Preface", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2017 IEEE.", "abstract": "Presents the introductory welcome message from the conference proceedings. May include the conference officers' congratulations to all involved with the conference event and publication of the proceedings record.", "date": "2017-10", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Piscataway, NJ", "pagerange": "xiii", "id_number": "CaltechAUTHORS:20191118-123714214", "isbn": "978-1-5386-3464-6", "book_title": "2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191118-123714214", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1109/FOCS.2017.5", "resource_type": "book_section", "pub_year": "2017", "author_list": "Umans, Chris" }, { "id": "https://authors.library.caltech.edu/records/gmfgg-yjb63", "eprint_id": 99871, "eprint_status": "archive", "datestamp": "2023-08-19 04:32:30", "lastmod": "2024-01-14 22:02:14", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hsu-Chloe Ching-Yun", "name": { "family": "Hsu", "given": "Chloe Ching-Yun" }, "orcid": "0000-0002-7743-3168" }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "On Multidimensional and Monotone k-SUM", "ispublished": "unpub", "full_text_status": "public", "keywords": "3SUM, kSUM, monotone 3SUM, strong 3SUM conjecture", "note": "\u00a9 2017 Chloe Ching-Yun Hsu and Christopher Umans; licensed under Creative Commons License CC-BY. \n\nSupported by NSF grant CCF-1423544 and a Simons Foundation Investigator grant.\n\nPublished - LIPIcs-MFCS-2017-50.pdf
", "abstract": "The well-known k-SUM conjecture is that integer k-SUM requires time \u03a9(n^([k/2]-o(1)). Recent work has studied multidimensional k-SUM in F^d_p, where the best known algorithm takes time O(n^([k/2]). Bhattacharyya et al. [ICS 2011] proved a min(2^(\u03a9(d)), n^(\u03a9(k)) lower bound for k-SUM in F^d_p under the Exponential Time Hypothesis. We give a more refined lower bound under the standard k-SUM conjecture: for sufficiently large p, k-SUM in F^d_p\nrequires time \u03a9(n^(k/2-o(1)) if k is even, and \u03a9(n^([k/2]-2k log k/log p \u2013o(1) if k is odd. For a special case of the multidimensional problem, bounded monotone d-dimensional 3SUM, Chan and Lewenstein [STOC 2015] gave a surprising O(n^(2\u22122/(d+13))) algorithm using additive combinatorics. We show this algorithm is essentially optimal. To be more precise, bounded monotone d-dimensional 3SUM requires time \u03a9(n^(2\u22124/d\u2212o(1))) under the standard 3SUM conjecture, and time \u03a9(n^(2\u22122/d\u2212o(1))) under the so-called strong 3SUM conjecture. Thus, even though one might hope to further exploit the structural advantage of monotonicity, no substantial improvements beyond those obtained by Chan and Lewenstein are possible for bounded monotone d-dimensional 3SUM.", "date": "2017-08", "date_type": "published", "publisher": "Dagstuhl Publishing", "place_of_pub": "Wadern, Germany", "pagerange": "Art. No. 50", "id_number": "CaltechAUTHORS:20191115-144427087", "isbn": "9783959770460", "book_title": "42nd International Symposium on Mathematical Foundations of Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191115-144427087", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-1423544" }, { "agency": "Simons Foundation" } ] }, "contributors": { "items": [ { "id": "Larsen-K-G", "name": { "family": "Larsen", "given": "Kim G." } }, { "id": "Bodlaender-H-L", "name": { "family": "Bodlaender", "given": "Hans L." } }, { "id": "Raskin-J-F", "name": { "family": "Raskin", "given": "Jean-Francois" } } ] }, "doi": "10.4230/LIPIcs.MFCS.2017.50", "primary_object": { "basename": "LIPIcs-MFCS-2017-50.pdf", "url": "https://authors.library.caltech.edu/records/gmfgg-yjb63/files/LIPIcs-MFCS-2017-50.pdf" }, "resource_type": "book_section", "pub_year": "2017", "author_list": "Hsu, Chloe Ching-Yun and Umans, Chris" }, { "id": "https://authors.library.caltech.edu/records/26zjc-te926", "eprint_id": 78787, "eprint_status": "archive", "datestamp": "2023-08-21 21:09:34", "lastmod": "2023-10-23 15:51:10", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hoza-W-M", "name": { "family": "Hoza", "given": "William M." } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "Targeted pseudorandom generators, simulation advice generators, and derandomizing logspace", "ispublished": "unpub", "full_text_status": "public", "keywords": "Theory of computation \u2192 Pseudorandomness and derandomization; Complexity classes; pseudorandom generators, derandomization, space complexity", "note": "\u00a9 2017 ACM. \n\nThe first author is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1610403. The second author is supported by National Science Foundation Grant No. CCF-1423544 and by a Simons Investigator grant.\n\nSubmitted - 1610.01199.pdf
", "abstract": "Assume that for every derandomization result for logspace algorithms, there is a pseudorandom generator strong enough to nearly recover the derandomization by iterating over all seeds and taking a majority vote. We prove under a precise version of this assumption that BPL \u2286 \u22c2_(\u03b1>0) DSPACE(log^(1 +\u03b1) n). \n\nWe strengthen the theorem to an equivalence by considering two generalizations of the concept of a pseudorandom generator against logspace. A targeted pseudorandom generator against logspace takes as input a short uniform random seed and a finite automaton; it outputs a long bitstring that looks random to that particular automaton. A simulation advice generator for logspace stretches a small uniform random seed into a long advice string; the requirement is that there is some logspace algorithm that, given a finite automaton and this advice string, simulates the automaton reading a long uniform random input. We prove that \u22c2_(\u03b1>0) prBPSPACE(log^(1+\u03b1)n) = \u22c2_(\u03b1>0)prDSPACE(log^(1+\u03b1)n) if and only if for every targeted pseudorandom generator against logspace, there is a simulation advice generator for logspace with similar parameters. \n\nFinally, we observe that in a certain uniform setting (namely, if we only worry about sequences of automata that can be generated in logspace), targeted pseudorandom generators against logspace can be transformed into simulation advice generators with similar parameters.", "date": "2017-06", "date_type": "published", "publisher": "ACM", "place_of_pub": "New York, NY", "pagerange": "629-640", "id_number": "CaltechAUTHORS:20170705-173204335", "isbn": "978-1-4503-4528-6", "book_title": "Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170705-173204335", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DGE-1610403" }, { "agency": "NSF", "grant_number": "CCF-1423544" }, { "agency": "Simons Foundation" } ] }, "doi": "10.1145/3055399.3055414", "primary_object": { "basename": "1610.01199.pdf", "url": "https://authors.library.caltech.edu/records/26zjc-te926/files/1610.01199.pdf" }, "resource_type": "book_section", "pub_year": "2017", "author_list": "Hoza, William M. and Umans, Chris" }, { "id": "https://authors.library.caltech.edu/records/zs1x3-06335", "eprint_id": 99906, "eprint_status": "archive", "datestamp": "2023-08-20 13:40:35", "lastmod": "2024-01-14 22:02:20", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Fefferman-B", "name": { "family": "Fefferman", "given": "Bill" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "On the Power of Quantum Fourier Sampling", "ispublished": "unpub", "full_text_status": "public", "keywords": "Quantum Complexity Theory, Sampling Complexity", "note": "\u00a9 2016 William Fefferman and Christopher Umans; licensed under Creative Commons License CC-BY. \n\nBF was supported by NSF CCF-1423544, BSF grant 2010120 and the Department of Defense. \n\nCU was supported by NSF CCF-1423544 and BSF grant 2010120.\n\nPublished - LIPIcs-TQC-2016-1.pdf
Submitted - 1507.05592.pdf
", "abstract": "A line of work initiated by Terhal and DiVincenzo [Terhal/DiVincenzo, arXiv, 2002] and Bremner, Jozsa, and Shepherd [Bremner/Jozsa/Sheperd, Proc. Royal Soc. A, 2010], shows that restricted classes of quantum computation can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer, unless the PH collapses. Aaronson and Arkhipov [Aaronson/Arkhipov, J. Theory of Comp., 2013] take this further by considering a distribution that can be sampled efficiently by linear optical quantum computation, that under two feasible conjectures, cannot even be approximately sampled within bounded total variation distance, unless the PH collapses. In this work we use Quantum Fourier Sampling to construct a class of distributions that can be sampled exactly by a quantum computer. We then argue that these distributions cannot be approximately sampled classically, unless the PH collapses, under variants of the Aaronson-Arkhipov conjectures. In particular, we show a general class of quantumly sampleable distributions each of which is based on an \"Efficiently Specifiable\" polynomial, for which a classical approximate sampler implies an average-case approximation. This class of polynomials contains the Permanent but also includes, for example, the Hamiltonian Cycle polynomial, as well as many other familiar #P-hard polynomials. Since our distribution likely requires the full power of universal quantum computation, while the Aaronson-Arkhipov distribution uses only linear optical quantum computation with noninteracting bosons, why is our result interesting? We can think of at least three reasons: 1. Since the conjectures required in [Aaronson/Arkhipov, J. Theory of Comp., 2013] have not yet been proven, it seems worthwhile to weaken them as much as possible. We do this in two ways, by weakening both conjectures to apply to any \"Efficiently Specifiable\" polynomial, and by weakening the so-called Anti-Concentration conjecture so that it need only hold for one distribution in a broad class of distributions. 2. Our construction can be understood without any knowledge of linear optics. While this may be a disadvantage for experimentalists, in our opinion it results in a very clean and simple exposition that may be more immediately accessible to computer scientists. 3. It is extremely common for quantum computations to employ \"Quantum Fourier Sampling\" in the following way: first apply a classically efficient function to a uniform superposition of inputs, then apply a Quantum Fourier Transform followed by a measurement. Our distributions are obtained in exactly this way, where the classically efficient function is related to a (presumed) hard polynomial. Establishing rigorously a robust sense in which the central primitive of Quantum Fourier Sampling is classically hard seems a worthwhile goal in itself.", "date": "2016-09", "date_type": "published", "publisher": "Dagstuhl Publishing", "place_of_pub": "Wadern, Germany", "pagerange": "Art. No. 1", "id_number": "CaltechAUTHORS:20191118-141357870", "isbn": "9783959770194", "book_title": "11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191118-141357870", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-1423544" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2010120" }, { "agency": "Department of Defense" } ] }, "contributors": { "items": [ { "id": "Broadbent-A", "name": { "family": "Broadbent", "given": "Anne" } } ] }, "doi": "10.4230/LIPIcs.TQC.2016.1", "primary_object": { "basename": "1507.05592.pdf", "url": "https://authors.library.caltech.edu/records/zs1x3-06335/files/1507.05592.pdf" }, "related_objects": [ { "basename": "LIPIcs-TQC-2016-1.pdf", "url": "https://authors.library.caltech.edu/records/zs1x3-06335/files/LIPIcs-TQC-2016-1.pdf" } ], "resource_type": "book_section", "pub_year": "2016", "author_list": "Fefferman, Bill and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/3ganf-61z78", "eprint_id": 99898, "eprint_status": "archive", "datestamp": "2023-08-20 13:13:54", "lastmod": "2024-01-14 22:02:16", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Guo-Zeyu", "name": { "family": "Guo", "given": "Zeyu" }, "orcid": "0000-0001-7893-4346" }, { "id": "Narayanan-A-K", "name": { "family": "Narayanan", "given": "Anand Kumar" }, "orcid": "0000-0002-0106-030X" }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields", "ispublished": "unpub", "full_text_status": "public", "keywords": "Algorithms, Complexity, Finite Fields, Polynomials, Factorization", "note": "\u00a9 2016 Zeyu Guo, Anand Kumar Narayanan and Chris Umans; licensed under Creative Commons License CC-BY. \n\nThe authors were supported by NSF grant CCF 1423544 and a Simons Foundation Investigator grant.\n\nPublished - LIPIcs-MFCS-2016-47.pdf
Submitted - 1606.04592.pdf
", "abstract": "The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes O(n^(3/2) log q+n log\u00b2 q time to factor polynomials of degree n over the finite field F_q with q elements. A significant open problem is if the 3/2 exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than 3/2 would yield an algorithm for polynomial factorization with exponent better than 3/2.", "date": "2016-08", "date_type": "published", "publisher": "Dagstuhl Publishing", "place_of_pub": "Wadern, Germany", "pagerange": "Art. No. 47", "id_number": "CaltechAUTHORS:20191118-103356388", "isbn": "9783959770163", "book_title": "41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191118-103356388", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-1423544" }, { "agency": "Simons Foundation" } ] }, "contributors": { "items": [ { "id": "Faliszewski-P", "name": { "family": "Faliszewski", "given": "Piotr" } }, { "id": "Muscholl-A", "name": { "family": "Muscholl", "given": "Anca" } }, { "id": "Niedermeier-R", "name": { "family": "Niedermeier", "given": "Rolf" } } ] }, "doi": "10.4230/LIPIcs.MFCS.2016.47", "primary_object": { "basename": "1606.04592.pdf", "url": "https://authors.library.caltech.edu/records/3ganf-61z78/files/1606.04592.pdf" }, "related_objects": [ { "basename": "LIPIcs-MFCS-2016-47.pdf", "url": "https://authors.library.caltech.edu/records/3ganf-61z78/files/LIPIcs-MFCS-2016-47.pdf" } ], "resource_type": "book_section", "pub_year": "2016", "author_list": "Guo, Zeyu; Narayanan, Anand Kumar; et el." }, { "id": "https://authors.library.caltech.edu/records/602cs-ph514", "eprint_id": 70326, "eprint_status": "archive", "datestamp": "2023-08-19 14:14:58", "lastmod": "2023-10-20 22:10:22", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Cohn-H", "name": { "family": "Cohn", "given": "Henry" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Fast matrix multiplication using coherent configurations", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2013 SIAM. \n\n[R]esearch supported by NSF grants CCF-0846991 and CCF-1116111 and BSF grant 2010120.\n\nPublished - 1_2E9781611973105_2E77.pdf
Submitted - 1207.6528.pdf
", "abstract": "We introduce a relaxation of the notion of tensor rank, called s-rank, and show that upper bounds on the s-rank of the matrix multiplication tensor imply upper bounds on the ordinary rank. In particular, if the \"s-rank exponent of matrix multiplication\" equals 2, then \u03c9 = 2. This connection between the s-rank exponent and the ordinary exponent enables us to significantly generalize the group-theoretic approach of Cohn and Umans, from group algebras to general algebras. Embedding matrix multiplication into general algebra multiplication yields bounds on s-rank (not ordinary rank) and, prior to this paper, that had been a barrier to working with general algebras.\n\nWe identify adjacency algebras of coherent configurations as a promising family of algebras in the generalized framework. Coherent configurations are combinatorial objects that generalize groups and group actions; adjacency algebras are the analogue of group algebras and retain many of their important features. As with groups, coherent configurations support matrix multiplication when a natural combinatorial condition is satisfied, involving triangles of points in their underlying geometry.\n\nFinally, we prove a closure property involving symmetric powers of adjacency algebras, which enables us to prove nontrivial bounds on \u03c9 using commutative coherent configurations and suggests that commutative coherent configurations may be sufficient to prove \u03c9 = 2. Altogether, our results show that bounds on \u03c9 can be established by embedding large matrix multiplication instances into small commutative coherent configurations.", "date": "2013-01", "date_type": "published", "publisher": "SIAM", "place_of_pub": "Philadelphia, PA", "pagerange": "1074-1087", "id_number": "CaltechAUTHORS:20160913-165513831", "isbn": "978-1-61197-251-1", "book_title": "Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160913-165513831", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-0846991" }, { "agency": "NSF", "grant_number": "CCF-1116111" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2010120" } ] }, "doi": "10.1137/1.9781611973105.77", "primary_object": { "basename": "1207.6528.pdf", "url": "https://authors.library.caltech.edu/records/602cs-ph514/files/1207.6528.pdf" }, "related_objects": [ { "basename": "1_2E9781611973105_2E77.pdf", "url": "https://authors.library.caltech.edu/records/602cs-ph514/files/1_2E9781611973105_2E77.pdf" } ], "resource_type": "book_section", "pub_year": "2013", "author_list": "Cohn, Henry and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/9jqzm-19e48", "eprint_id": 100072, "eprint_status": "archive", "datestamp": "2023-08-19 11:15:27", "lastmod": "2023-10-18 19:02:37", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ta-Shma-A", "name": { "family": "Ta-Shma", "given": "Amnon" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Better Condensers and New Extractors from Parvaresh-Vardy Codes", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "randomness extractors, Parvaresh-Vardy codes, condensers", "note": "\u00a9 2012 IEEE. \n\nAmnon Ta-Shma was supported by Grant No. 2010120 from the United States-Israel Binational Science Foundation (BSF), and by Grant No. 1090/10 from the Israel Science foundation (ISF). Christopher Umans was supported by NSF CCF-0846991, CCF-1116111 and BSF grant 2010120. We thank Zeev Dvir, Ariel Gabizon, Swastik Kopparty, and Ronen Shaltiel for useful discussions. Thanks for Zeev Dvir for confirming that the extractors of [3] work with smaller \u03b5 than claimed in the paper.", "abstract": "We give a new construction of condensers based on Parvaresh-Vardy codes [1]. Our condensers have entropy rate (1-\u03b1) for subconstant \u03b1 (in contrast to [2] which required constant \u03b1) and suffer only sublinear entropy loss. Known extractors can be applied to the output to extract all but a subconstant fraction of the minentropy. The resulting (k, \u03b5) extractor E : {0, 1}^n \u00d7 {0, 1}^d \u2192 {0, 1}^m has output length m = (1- \u03b1)k with \u03b1 = 1/poly log(n), and seed length d = O(log n), when \u03b5 \u2265 \u00bd^(log\u03b2 n) for any constant \u00df < 1. Thus we achieve the same \"world-record\" extractor parameters as [3], with a more direct construction.", "date": "2012-06", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Piscataway, NJ", "pagerange": "309-315", "id_number": "CaltechAUTHORS:20191126-140535982", "isbn": "9780769547084", "book_title": "2012 IEEE 27th Conference on Computational Complexity", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-140535982", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2010120" }, { "agency": "Israel Science Foundation", "grant_number": "1090/10" }, { "agency": "NSF", "grant_number": "CCF-0846991" }, { "agency": "NSF", "grant_number": "CCF-1116111" } ] }, "doi": "10.1109/ccc.2012.25", "resource_type": "book_section", "pub_year": "2012", "author_list": "Ta-Shma, Amnon and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/5jdxk-df270", "eprint_id": 39214, "eprint_status": "archive", "datestamp": "2023-08-19 11:12:28", "lastmod": "2024-01-13 06:02:13", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Alon-Noga", "name": { "family": "Alon", "given": "Noga" }, "orcid": "0000-0003-1332-4883" }, { "id": "Shpilka-A", "name": { "family": "Shpilka", "given": "Amir" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "On Sunflowers and Matrix Multiplication", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Matrix Multiplication, Sunflower Conjecture", "note": "\u00a9 2012 IEEE. We thank Adam Marcus for helpful discussions in an early stage of this research. We thank Henry Cohn, Bobby Kleinberg, and Bal\u00e1zs Szegedy for help in formulating the weak sunflower conjecture in Z^n_D, and for useful discussions. Thanks to Gil Kalai for his comments and questions. The research of Noga Alon was supported in part by an ERC Advanced grant and by NSF grant No. DMS-0835373. \n\nAmir Shpilka has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement number 257575. The research of Christopher Umans was supported by NSF CCF-0846991, CCF-1116111 and BSF grant 2010120.", "abstract": "We present several variants of the sunflower conjecture of Erdos and Rado [ER60] and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Wino grad [CW90] and Cohn et al [CKSU05] regarding possible approaches for obtaining fast matrix multiplication algorithms. Specifically, we show that the Erdos-Rado sunflower conjecture (if true) implies a negative answer to the \"no three disjoint equivoluminous subsets\" question of Coppersmith and Wino grad [CW90]; we also formulate a \"multicolored'' sunflower conjecture in Z^n_3 and show that (if true) it implies a negative answer to the \"strong USP\" conjecture of [CKSU05] (although it does not seem to impact a second conjecture in [CKSU05] or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith-Wino grad conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in Z^n_3 is a strengthening of the well-known (ordinary) sunflower conjecture in Z^n_3, and we show via our connection that a construction from [CKSU05] yields a lower bound of (2.51...)^n on the size of the largest multicolored 3-sunflower-free set, which beats the current best known lower bound of (2.21...)^n [Edel04] on the size of the largest 3-sunflower-free set in Z^n_3.", "date": "2012-06", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Piscataway, NJ", "pagerange": "214-223", "id_number": "CaltechAUTHORS:20130703-110037886", "isbn": "978-1-4673-1663-7", "book_title": "27th Annual IEEE Conference on Computational Complexity (CCC)", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130703-110037886", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0835373" }, { "agency": "European Research Council (ERC)", "grant_number": "257575" }, { "agency": "NSF", "grant_number": "CCF-0846991" }, { "agency": "NSF", "grant_number": "CCF-1116111" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2010120" } ] }, "doi": "10.1109/CCC.2012.26", "resource_type": "book_section", "pub_year": "2012", "author_list": "Alon, Noga; Shpilka, Amir; et el." }, { "id": "https://authors.library.caltech.edu/records/0v97s-fy668", "eprint_id": 31588, "eprint_status": "archive", "datestamp": "2023-08-19 09:25:58", "lastmod": "2023-10-17 18:41:59", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Fefferman-B", "name": { "family": "Fefferman", "given": "Bill" } }, { "id": "Shaltiel-R", "name": { "family": "Shaltiel", "given": "Ronen" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } }, { "id": "Viola-E", "name": { "family": "Viola", "given": "Emanuele" } } ] }, "title": "On beating the hybrid argument", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2012 ACM. Supported by IQI. Supported by BSF grant 2010120, ISF grants 686/07, 864/11 and ERC starting grant 279559. Supported by NSF CCF-0846991, CCF-1116111 and BSF\ngrant 2010120. Supported by NSF grant CCF-0845003.", "abstract": "The hybrid argument allows one to relate the distinguishability of a distribution (from uniform) to the predictability of individual bits given a prefix. The argument incurs a loss of a factor k equal to the bit-length of the distributions: \u03b5-distinguishability implies \u03b5/k-predictability. This paper studies the consequences of avoiding this loss - what we call \"beating the hybrid argument\" -- and develops new proof techniques that circumvent the loss in certain natural settings. Specifically, we obtain the following results:\n1. We give an instantiation of the Nisan-Wigderson generator (JCSS '94) that can be broken by quantum computers, and that is o(1)-unpredictable against AC^0. We conjecture that this generator indeed fools AC^0. Our conjecture implies the existence of an oracle relative to which BQP is not in the PH, a longstanding open problem.\n2. We show that the \"INW\" generator by Impagliazzo, Nisan, and Wigderson (STOC '94) with seed length O(log n log log n) produces a distribution that is 1/log n-unpredictable against poly-logarithmic width (general) read-once oblivious branching programs. Obtaining such generators where the output is indistinguishable from uniform is a longstanding open problem.\n3. We identify a property of functions f, \"resamplability,\" that allows us to beat the hybrid argument when arguing indistinguishability of\n[EQUATION] from uniform. This gives new pseudorandom generators for classes such as AC^0[p] with a stretch that, despite being sub-linear, is the largest known. We view this as a first step towards beating the hybrid argument in the analysis of the Nisan-Wigderson generator (which applies G_f^\u2297k on correlated x_1,...,x_k) and proving the conjecture in the first item.", "date": "2012-01", "date_type": "published", "publisher": "Association for Computing Machinery (ACM)", "place_of_pub": "New York, NY", "pagerange": "468-483", "id_number": "CaltechAUTHORS:20120522-093532704", "isbn": "978-1-4503-1115-1", "book_title": "Proceedings of the 3rd Innovations in Theoretical Computer Science Conference", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120522-093532704", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "IQI" }, { "agency": "BSF", "grant_number": "2010120" }, { "agency": "ISF", "grant_number": "686/07" }, { "agency": "ISF", "grant_number": "864/11" }, { "agency": "ERC Starting Grant", "grant_number": "279559" }, { "agency": "NSF", "grant_number": "CCF-0846991" }, { "agency": "NSF", "grant_number": "CCF-1116111" }, { "agency": "BSF", "grant_number": "2010120" }, { "agency": "NSF", "grant_number": "CCF-0845003" } ] }, "doi": "10.1145/2090236.2090273", "resource_type": "book_section", "pub_year": "2012", "author_list": "Fefferman, Bill; Shaltiel, Ronen; et el." }, { "id": "https://authors.library.caltech.edu/records/6ge7e-jtr53", "eprint_id": 19608, "eprint_status": "archive", "datestamp": "2023-08-19 00:58:54", "lastmod": "2024-01-12 23:59:38", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Buchfuhrer-Dave", "name": { "family": "Buchfuhrer", "given": "Dave" } }, { "id": "Dughmi-Shaddin", "name": { "family": "Dughmi", "given": "Shaddin" } }, { "id": "Fu-Hu", "name": { "family": "Fu", "given": "Hu" } }, { "id": "Kleinberg-Robert", "name": { "family": "Kleinberg", "given": "Robert" } }, { "id": "Mossel-E", "name": { "family": "Mossel", "given": "Elchanan" }, "orcid": "0000-0001-7812-7886" }, { "id": "Papadimitriou-Christos-H", "name": { "family": "Papadimitriou", "given": "Christos" } }, { "id": "Schapira-Michael", "name": { "family": "Schapira", "given": "Michael" } }, { "id": "Singer-Yaron", "name": { "family": "Singer", "given": "Yaron" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "Inapproximability for VCG-Based Combinatorial Auctions", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2010 SIAM.", "abstract": "The existence of incentive-compatible, computationally efficient\nmechanisms for combinatorial auctions with\ngood approximation ratios is the paradigmatic problem\nin algorithmic mechanism design. It is believed that, in\nmany cases, good approximations for combinatorial auctions\nmay be unattainable due to an inherent clash between\ntruthfulness and computational efficiency. In this\npaper, we prove the first computational-complexity inapproximability\nresults for incentive-compatible mechanisms\nfor combinatorial auctions. Our results are tight,\nhold for the important class of VCG-based mechanisms,\nand are based on the complexity assumption that NP\nhas no polynomial-size circuits. We show two different\ntechniques to obtain such lower bounds: one for deterministic mechanisms that attains optimal dependence\non the number of players and number of items, and one\nthat also applies to a class of randomized mechanisms\nand attains optimal dependence on the number of players.\nBoth techniques are based on novel VC dimension\nmachinery.", "date": "2010", "date_type": "published", "publisher": "Association for Computing Machinery", "place_of_pub": "New York, NY", "pagerange": "518-536", "id_number": "CaltechAUTHORS:20100824-073813316", "isbn": "978-0-898717-01-3", "book_title": "Proceedings of the Twenty-first Annual ACM-SIAM Symposium on Discrete Algorithms", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100824-073813316", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "contributors": { "items": [ { "id": "Charikar-M", "name": { "family": "Charikar", "given": "Moses" } } ] }, "resource_type": "book_section", "pub_year": "2010", "author_list": "Buchfuhrer, Dave; Dughmi, Shaddin; et el." }, { "id": "https://authors.library.caltech.edu/records/xawwq-q5d24", "eprint_id": 100084, "eprint_status": "archive", "datestamp": "2023-08-19 00:15:20", "lastmod": "2023-10-18 19:03:06", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Agrawal-M", "name": { "family": "Agrawal", "given": "Manindra" } }, { "id": "Fortnow-L", "name": { "family": "Fortnow", "given": "Lance" } }, { "id": "Thierauf-T", "name": { "family": "Thierauf", "given": "Thomas" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Algebraic Methods in Computational Complexity", "ispublished": "unpub", "full_text_status": "public", "keywords": "Computational Complexity, Algebra", "note": "\u00a9 2009 Dagstuhl Publishing.\n\nPublished - 09421_abstracts_collection.2418.pdf
", "abstract": "From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 \"Algebraic Methods in Computational Complexity \" was held in Schloss Dagstuhl-Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.", "date": "2009-10", "date_type": "published", "publisher": "Dagstuhl Publishing", "id_number": "CaltechAUTHORS:20191127-075729203", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191127-075729203", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "primary_object": { "basename": "09421_abstracts_collection.2418.pdf", "url": "https://authors.library.caltech.edu/records/xawwq-q5d24/files/09421_abstracts_collection.2418.pdf" }, "resource_type": "book_section", "pub_year": "2009", "author_list": "Agrawal, Manindra; Fortnow, Lance; et el." }, { "id": "https://authors.library.caltech.edu/records/dkhwk-f5y53", "eprint_id": 100094, "eprint_status": "archive", "datestamp": "2023-08-19 00:15:25", "lastmod": "2023-10-18 19:03:52", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Agrawal-M", "name": { "family": "Agrawal", "given": "Manindra" } }, { "id": "Fortnow-L", "name": { "family": "Fortnow", "given": "Lance" } }, { "id": "Thierauf-T", "name": { "family": "Thierauf", "given": "Thomas" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "Algebraic Methods in Computational Complexity", "ispublished": "unpub", "full_text_status": "public", "keywords": "Computational Complexity, Algebra", "note": "\u00a9 2009 Dagstuhl Publishing.\n\nPublished - 09421.SWM.ExtendedAbstract.2410.pdf
", "abstract": "The seminar brought together more than 50 researchers covering a wide spectrum of complexity theory. The focus on algebraic methods showed once again the great importance of algebraic techniques for theoretical computer science. We had almost 30 talks, most of them about 40 minutes leaving ample room for discussions. We also had a much appreciated open problem session. The talks ranged over a broad assortment of subjects with the underlying theme of using algebraic techniques. It was very fruitful and has hopefully initiated new directions in research. Several participants specifically mentioned that they appreciated the particular focus on a common class of techniques (rather than end results) as a unifying theme of the workshop. We look forward to our next meeting!", "date": "2009-10", "date_type": "published", "publisher": "Dagstuhl Publishing", "id_number": "CaltechAUTHORS:20191127-093701543", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191127-093701543", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "primary_object": { "basename": "09421.SWM.ExtendedAbstract.2410.pdf", "url": "https://authors.library.caltech.edu/records/dkhwk-f5y53/files/09421.SWM.ExtendedAbstract.2410.pdf" }, "resource_type": "book_section", "pub_year": "2009", "author_list": "Agrawal, Manindra; Fortnow, Lance; et el." }, { "id": "https://authors.library.caltech.edu/records/rg0vv-3zb81", "eprint_id": 18921, "eprint_status": "archive", "datestamp": "2023-08-19 00:13:51", "lastmod": "2024-01-12 23:41:05", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kalyanaraman-S", "name": { "family": "Kalyanaraman", "given": "Shankar" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "The Complexity of Rationalizing Network Formation", "ispublished": "unpub", "full_text_status": "public", "keywords": "network formation games, rationalization, Jackson-\nWolinsky model, Inequality-SAT, hardness of approximation", "note": "\u00a9 2009 IEEE.\n\nIssue Date: 25-27 Oct. 2009; Date of Current Version: 25 March 2010.\n\nWe thank Prasad Raghavendra for a helpful conversation,\nand the FOCS referees for their comments.\n\nPublished - Kalyanaraman2009p1051346Th_Annual_Ieee_Symposium_On_Foundations_Of_Computer_Science_Proceedings.pdf
", "abstract": "We study the complexity of rationalizing network formation. In this problem we fix an underlying model describing how selfish parties (the vertices) produce a graph by making individual decisions to form or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and we observe a set of \"snapshots\" of graphs that are assumed to be stable. From this we would like to infer some unobserved data about the system: edge prices, or how much each vertex values short paths to each other vertex. We study two rationalization problems arising from the network formation model of Jackson and Wolinsky [14]. When the goal is to infer edge prices, we observe that the rationalization problem is easy. The problem remains easy even when rationalizing prices do not exist and we instead wish to find prices that maximize the stability of the system. In contrast, when the edge prices are given and the goal is instead to infer valuations of each vertex by each other vertex, we prove that the rationalization problem becomes NP-hard. Our proof exposes a close connection between rationalization problems and the Inequality-SAT (I-SAT) problem. Finally and most significantly, we prove that an approximation version of this NP-complete rationalization problem is NP-hard to approximate to within better than a 1/2 ratio. This shows that the trivial algorithm of setting everyone's valuations to infinity (which rationalizes all the edges present in the input graphs) or to zero (which rationalizes all the non-edges present in the input graphs) is the best possible assuming P \u2260 NP To do this we prove a tight (1/2 + \u03b4) -approximation hardness for a variant of I-SAT in which all coefficients are non-negative. This in turn follows from a tight hardness result for MAX-LlN_(R_+) (linear equations over the reals, with non-negative coefficients), which we prove by a (non-trivial) modification of the recent result of Guruswami and Raghavendra [10] which achieved tight hardness for this problem without the non-negativity constraint. Our technical contributions regarding the hardness of I-SAT and MAX-LIN_(R_+) may be of independent interest, given the generality of these problems", "date": "2009-10", "date_type": "published", "publisher": "IEEE", "pagerange": "485-494", "id_number": "CaltechAUTHORS:20100707-095613286", "isbn": "978-1-4244-5116-6", "book_title": "50th Annual IEEE Symposium on Foundations of Computer Science, 2009", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100707-095613286", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "other_numbering_system": { "items": [ { "id": "11207116", "name": "INSPEC Accession Number" } ] }, "doi": "10.1109/FOCS.2009.48", "primary_object": { "basename": "Kalyanaraman2009p1051346Th_Annual_Ieee_Symposium_On_Foundations_Of_Computer_Science_Proceedings.pdf", "url": "https://authors.library.caltech.edu/records/rg0vv-3zb81/files/Kalyanaraman2009p1051346Th_Annual_Ieee_Symposium_On_Foundations_Of_Computer_Science_Proceedings.pdf" }, "resource_type": "book_section", "pub_year": "2009", "author_list": "Kalyanaraman, Shankar and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/8w7ns-tq041", "eprint_id": 100080, "eprint_status": "archive", "datestamp": "2023-08-20 00:10:57", "lastmod": "2024-01-14 22:02:58", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kalyanaraman-S", "name": { "family": "Kalyanaraman", "given": "Shankar" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "The Complexity of Rationalizing Matchings", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2008 Springer-Verlag Berlin Heidelberg. \n\nSupported by NSF CCF-0346991, NSF CCF-0830787, BSF 2004329 and a Graduate Research Fellowship from the Social and Information Sciences Laboratory (SISL) at Caltech. \n\nSupported by NSF CCF-0346991, NSF CCF-0830787, BSF 2004329, a Sloan Research Fellowship, and an Okawa Foundation research grant. \n\nWe are indebted to Federico Echenique for numerous invaluable discussions and for getting us started on this work.", "abstract": "Given a set of observed economic choices, can one infer preferences and/or utility functions for the players that are consistent with the data? Questions of this type are called rationalization or revealed preference problems in the economic literature, and are the subject of a rich body of work.\n\nFrom the computer science perspective, it is natural to study the complexity of rationalization in various scenarios. We consider a class of rationalization problems in which the economic data is expressed by a collection of matchings, and the question is whether there exist preference orderings for the nodes under which all the matchings are stable.\n\nWe show that the rationalization problem for one-one matchings is NP-complete. We propose two natural notions of approximation, and show that the problem is hard to approximate to within a constant factor, under both. On the positive side, we describe a simple algorithm that achieves a 3/4 approximation ratio for one of these approximation notions. We also prove similar results for a version of many-one matching.", "date": "2008-12-10", "date_type": "published", "publisher": "Springer", "place_of_pub": "Berlin", "pagerange": "171-182", "id_number": "CaltechAUTHORS:20191126-155702845", "isbn": "978-3-540-92181-3", "book_title": "Algorithms and Computation", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-155702845", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "NSF", "grant_number": "CCF-0830787" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" }, { "agency": "Caltech Social and Information Sciences Laboratory" }, { "agency": "NSF Graduate Research Fellowship" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Okawa Foundation" } ] }, "contributors": { "items": [ { "id": "Hong-Seok-Hee", "name": { "family": "Hong", "given": "Seok-Hee" } }, { "id": "Nagamochi-Hiroshi", "name": { "family": "Nagamochi", "given": "Hiroshi" } }, { "id": "Fukunaga-Takuro", "name": { "family": "Fukunaga", "given": "Takuro" } } ] }, "doi": "10.1007/978-3-540-92182-0_18", "resource_type": "book_section", "pub_year": "2008", "author_list": "Kalyanaraman, Shankar and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/6bpbb-0gh88", "eprint_id": 100081, "eprint_status": "archive", "datestamp": "2023-08-19 23:44:40", "lastmod": "2023-10-18 19:02:58", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kedlaya-K-S", "name": { "family": "Kedlaya", "given": "Kiran S." } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Fast Modular Composition in any Characteristic", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2008 IEEE. \n\nSupported by NSF DMS-0545904 (CAREER) and a Sloan Research\nFellowship. \n\nSupported by NSF CCF-0346991, BSF 2004329, a Sloan Research\nFellowship, and an Okawa Foundation research grant. \n\nWe thank Swastik Kopparty and Madhu Sudan for some references mentioned in Section 4, and Ronald deWolf and the FOCS 2008 referees for helpful comments.\n\nPublished - 04690949.pdf
", "abstract": "We give an algorithm for modular composition of degree n univariate polynomials over a finite field F_q requiring n^(1 + o(1))log^(1 + o(1))q bit operations; this had earlier been achieved in characteristic n^(o(1)) by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over F_q requiring (n^(1.5 + o(1)) + n^(1 + o(1)) log q) log^(1 + o(1)) q bit operations, improving upon the methods of von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998). Our results also imply algorithms for irreducibility testing and computing minimal polynomials whose running times are best-possible, up to lower order terms.As in Umans (2008), we reduce modular composition to certain instances of multipoint evaluation of multivariate polynomials. We then give an algorithm that solves this problem optimally (up to lower order terms), in arbitrary characteristic. The main idea is to lift to characteristic 0, apply a small number of rounds of multimodular reduction, and finish with a small number of multidimensional FFTs. The final evaluations are then reconstructed using the Chinese Remainder Theorem. As a bonus, we obtain a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithm uses techniques which are commonly employed in practice, so it may be competitive for real problem sizes. This contrasts with previous asymptotically fast methods relying on fast matrix multiplication.", "date": "2008-10", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Piscataway, NJ", "pagerange": "146-155", "id_number": "CaltechAUTHORS:20191126-160438923", "isbn": "978-0-7695-3436-7", "book_title": "2008 49th Annual IEEE Symposium on Foundations of Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-160438923", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0545904" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" }, { "agency": "Okawa Foundation" } ] }, "doi": "10.1109/focs.2008.13", "primary_object": { "basename": "04690949.pdf", "url": "https://authors.library.caltech.edu/records/6bpbb-0gh88/files/04690949.pdf" }, "resource_type": "book_section", "pub_year": "2008", "author_list": "Kedlaya, Kiran S. and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/707p7-36d28", "eprint_id": 100095, "eprint_status": "archive", "datestamp": "2023-08-22 12:49:56", "lastmod": "2023-10-18 19:03:54", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kedlaya-K-S", "name": { "family": "Kedlaya", "given": "Kiran S." } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Fast polynomial factorization and modular composition", "ispublished": "unpub", "full_text_status": "public", "keywords": "Modular composition; polynomial factorization; multipoint evaluation; Chinese Remaindering", "note": "\u00a9 2008 Dagstuhl Publishing.\n\nThe material in this paper appeared in conferences as [Uma08] and [KU08].\n\nSupported by NSF DMS-0545904 (CAREER) and a Sloan Research Fellowship.\n\nSupported by NSF CCF-0346991 (CAREER), CCF-0830787, BSF 2004329, and a Sloan Research Fellowship.\n\nWe thank Henry Cohn, Joachim von zur Gathen, David Harvey, Erich Kaltofen, and Eyal Rozenman for\nuseful discussions, and \u00c9ric Schost for helpful comments on a draft of [Uma08]. We thank Swastik Kopparty\nand Madhu Sudan for some references mentioned in Section 5, and Ronald de Wolf and the FOCS 2008\nreferees for helpful comments on the conference paper [KU08]. Finally, we thank Madhu Sudan for hosting\na visit of the second author to MIT, which launched this collaboration.\n\nPublished - 08381.UmansChristopher.Paper.1777.pdf
", "abstract": "We obtain randomized algorithms for factoring degree n univariate polynomials over F_q requiring\nO(^(n1.5+o(1)) log^(1+o(1)) q + n^(1+o(1)) log^(2+o(1)) q) bit operations. When log q < n, this is asymptotically\nfaster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998));\nfor log q \u2265 n, it matches the asymptotic running time of the best known algorithms.\nThe improvements come from new algorithms for modular composition of degree n univariate polynomials,\nwhich is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields.\nThe best previous algorithms for modular composition use O(n^((\u03c9+1)/2)) field operations, where \u03c9 is the\nexponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent\nachieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)).\nWe show that modular composition and multipoint evaluation of multivariate polynomials are essentially\nequivalent, in the sense that an algorithm for one achieving exponent \u03b1 implies an algorithm\nfor the other with exponent \u03b1 + o(1), and vice versa. We then give two new algorithms that solve the\nproblem optimally (up to lower order terms): an algebraic algorithm for fields of characteristic at most\nn^(o(1)), and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by\nlifting to characteristic 0, applying a small number of rounds of multimodular reduction, and finishing\nwith a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese\nRemainder Theorem. As a bonus, this algorithm produces a very efficient data structure supporting\npolynomial evaluation queries, which is of independent interest.\nOur algorithms use techniques which are commonly employed in practice, so they may be competitive for real problem sizes. This contrasts with all previous subquadratic algorithsm for these problems, which rely on fast matrix multiplication.", "date": "2008-08-31", "date_type": "published", "publisher": "Dagstuhl Publishing", "id_number": "CaltechAUTHORS:20191127-094213132", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191127-094213132", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0545904" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "NSF", "grant_number": "CCF-0830787" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" } ] }, "primary_object": { "basename": "08381.UmansChristopher.Paper.1777.pdf", "url": "https://authors.library.caltech.edu/records/707p7-36d28/files/08381.UmansChristopher.Paper.1777.pdf" }, "resource_type": "book_section", "pub_year": "2008", "author_list": "Kedlaya, Kiran S. and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/w5tws-j0w81", "eprint_id": 99813, "eprint_status": "archive", "datestamp": "2023-08-22 12:44:04", "lastmod": "2024-01-14 22:02:10", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Buchfuhrer-D", "name": { "family": "Buchfuhrer", "given": "David" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "The Complexity of Boolean Formula Minimization", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Positive Instance; Boolean Formula; Negative Instance; Boolean Circuit; Equivalent Formula", "note": "\u00a9 2008 Springer-Verlag Berlin Heidelberg. \n\nSupported by NSF CCF-0346991 and BSF 2004329. \n\nSupported by NSF CCF-0346991, BSF 2004329, a Sloan Research Fellowship, and an Okawa Foundation research grant.", "abstract": "The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be \u03a3^P\u2082-complete and indeed appears as an open problem in Garey and Johnson [GJ79]. The depth-2 variant was only shown to be \u03a3^P\u2082-complete in 1998 [Uma98], and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth-k version is \u03a3^P\u2082-complete under Turing reductions for all k\u2009\u2265\u20093. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is \u03a3^P\u2082-complete under Turing reductions.", "date": "2008-08-12", "date_type": "published", "publisher": "Springer", "place_of_pub": "Berlin", "pagerange": "24-35", "id_number": "CaltechAUTHORS:20191112-131623581", "isbn": "978-3-540-70574-1", "book_title": "Automata, Languages and Programming", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191112-131623581", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Okawa Foundation" } ] }, "contributors": { "items": [ { "id": "Aceto-L", "name": { "family": "Aceto", "given": "Luca" } }, { "id": "Damg\u00e5rd-I", "name": { "family": "Damg\u00e5rd", "given": "Ivan" } }, { "id": "Goldberg-L-A", "name": { "family": "Goldberg", "given": "Leslie Ann" } }, { "id": "Halld\u00f3rsson-M-M", "name": { "family": "Halld\u00f3rsson", "given": "Magn\u00fas M." } }, { "id": "Ing\u00f3lfsd\u00f3ttir-A", "name": { "family": "Ing\u00f3lfsd\u00f3ttir", "given": "Anna" } }, { "id": "Walukiewicz-I", "name": { "family": "Walukiewicz", "given": "Igor" } } ] }, "doi": "10.1007/978-3-540-70575-8_3", "resource_type": "book_section", "pub_year": "2008", "author_list": "Buchfuhrer, David and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/cm9h5-57639", "eprint_id": 73193, "eprint_status": "archive", "datestamp": "2023-08-19 22:37:56", "lastmod": "2023-10-24 15:06:52", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Fast polynomial factorization and modular composition in small characteristic", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2008 ACM. \n\nSupported by NSF CCF-0346991, BSF 2004329, a Sloan Research Fellowship, and an Okawa Foundation research grant.", "abstract": "We obtain randomized algorithms for factoring degree n univariate polynomials over F_q that use O(n^(1.5 + o(1)) + n^(1 + o(1))log q) field operations, when the characteristic is at most n^(o(1)). When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for log q \u2265 n, it matches the asymptotic running time of the best known algorithms.\n\nThe improvements come from a new algorithm for modular composition of degree n univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use O(n^((\u03c9+ 1)/2)) field operations, where \u03c9is the exponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)).\n\nWe show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent in the sense that an algorithm for one achieving exponent \u03b1 implies an algorithm for the other with exponent \u03b1 + o(1), and vice versa. We then give a new algorithm that requires O(n^(1 + o(1))) field operations when the characteristic is at most n^(o(1)), which is optimal up to lower order terms.\n\nOur algorithms do not rely on fast matrix multiplication, in contrast to all previous subquadratic algorithms for these problems. The main operations are fast univariate polynomial arithmetic, multipoint evaluation, and interpolation, and consequently the algorithms could be feasible in practice.", "date": "2008-05", "date_type": "published", "publisher": "ACM", "place_of_pub": "New York, NY", "pagerange": "481-490", "id_number": "CaltechAUTHORS:20170103-171822942", "isbn": "978-1-60558-047-0", "book_title": "STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170103-171822942", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Okawa Foundation" } ] }, "contributors": { "items": [ { "id": "Ladner-R", "name": { "family": "Ladner", "given": "Richard" } } ] }, "doi": "10.1145/1374376.1374445", "resource_type": "book_section", "pub_year": "2008", "author_list": "Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/fvsxd-8yb55", "eprint_id": 98295, "eprint_status": "archive", "datestamp": "2023-08-22 10:01:10", "lastmod": "2024-01-14 21:55:09", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kalyanaraman-S", "name": { "family": "Kalyanaraman", "given": "Shankar" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Algorithms for Playing Games with Limited Randomness", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Nash Equilibrium; Mixed Strategy; Player Game; Online Algorithm; Congestion Game", "note": "\u00a9 Springer-Verlag Berlin Heidelberg 2007. \n\nThis research was supported by NSF grant CCF-0346991, BSF Grant 2004329, a Sloan Research Fellowship and an Okawa Foundation research grant.", "abstract": "We study multiplayer games in which the participants have access to only limited randomness. This constrains both the algorithms used to compute equilibria (they should use little or no randomness) as well as the mixed strategies that the participants are capable of playing (these should be sparse). We frame algorithmic questions that naturally arise in this setting, and resolve several of them. \n\nWe give deterministic algorithms that can be used to find sparse \u03b5-equilibria in zero-sum and non-zero-sum games, and a randomness-efficient method for playing repeated zero-sum games. These results apply ideas from derandomization (expander walks, and \u03b4-independent sample spaces) to the algorithms of Lipton, Markakis, and Mehta [LMM03], and the online algorithm of Freund and Schapire [FS99]. \n\nSubsequently, we consider a large class of games in which sparse equilibria are known to exist (and are therefore amenable to randomness-limited players), namely games of small rank. We give the first \"fixed-parameter\" algorithms for obtaining approximate equilibria in these games. For rank-k games, we give a deterministic algorithm to find (k\u2009+\u20091)-sparse \u03b5-equilibria in time polynomial in the input size n and some function f(k,1/\u03b5). In a similar setting Kannan and Theobald [KT07] gave an algorithm whose run-time is n^(O(k)). Our algorithm works for a slightly different notion of a game's \"rank,\" but is fixed parameter tractable in the above sense, and extends easily to the multi-player case.", "date": "2007-09-14", "date_type": "published", "publisher": "Springer Berlin Heidelberg", "place_of_pub": "Berlin, Heidelberg", "pagerange": "323-334", "id_number": "CaltechAUTHORS:20190828-102317126", "isbn": "9783540755197", "book_title": "Algorithms \u2013 ESA 2007", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190828-102317126", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Okawa Foundation" } ] }, "contributors": { "items": [ { "id": "Arge-L", "name": { "family": "Arge", "given": "Lars" } }, { "name": { "family": "Hoffmann", "given": "M." } }, { "id": "Welzl-E", "name": { "family": "Welzl", "given": "Emo" } } ] }, "doi": "10.1007/978-3-540-75520-3_30", "resource_type": "book_section", "pub_year": "2007", "author_list": "Kalyanaraman, Shankar and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/1dm9g-bnm76", "eprint_id": 72944, "eprint_status": "archive", "datestamp": "2023-08-19 20:22:30", "lastmod": "2023-10-23 23:25:51", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Shaltiel-R", "name": { "family": "Shaltiel", "given": "Ronen" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Low-end uniform hardness vs. randomness tradeoffs for AM", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Theory, Algorithms, Arthur-Merlin games, hardness vs. randomness tradeoff, derandomization, hitting-set generator", "note": "\u00a9 2007 ACM. \n\nSupported by BSF grant 2004329. \n\nSupported by NSF CCF-0346991, BSF 2004329, a Sloan Research Fellowship, and an Okawa Foundation research grant.", "abstract": "In 1998, Impagliazzo and Wigderson [18] proved a hardness vs. randomness tradeoff for BPP in the uniform setting,which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions by Trevisan and Vadhan [29] (in a slightly weaker setting). In 2003, Gutfreund,Shaltiel and Ta-Shma [11] proved a uniform hardness vs. randomness tradeoff for AM, but that result only worked on the \"high-end\" of possible hardness assumptions.\n\nIn this work, we give uniform hardness vs. randomness tradeoffs for AM that are near-optimal for the full range of possible hardness assumptions. Following [11], we do this by constructing a hitting-set-generator (HSG) for AM with \"resilient reconstruction.\" Our construction is a recursive variant of the Miltersen-Vinodchandran HSG [24], the only known HSG construction with this required property. The main new idea is to have the reconstruction procedure operate implicitly and locally on superpolynomially large objects, using tools from PCPs(low-degree testing, self-correction) together with a novel use of extractors that are built from Reed-Muller codes [28, 26] for a sort of locally-computable error-reduction. \n\nAs a consequence we obtain gap theorems for AM (and AM \u2229 coAM) that state, roughly, that either AM (or AM \u2229 coAM)protocols running in time t(n) can simulate all of EXP(\"Arthur-Merlin games are powerful\"), or else all of AM (or AM \u2229 coAM) can be simulated in nondeterministic time s(n) (\"Arthur-Merlin games can be derandomized\"), for a near-optimal relationship between t(n) and s(n). As in GST, the case of AM \u2229 coAM yields a particularly clean theorem that is of special interest due to the wide array of cryptographic and other problems that lie in this class.", "date": "2007-06", "date_type": "published", "publisher": "ACM", "place_of_pub": "New York, NY", "pagerange": "430-439", "id_number": "CaltechAUTHORS:20161219-151217993", "isbn": "978-1-59593-631-8", "book_title": "STOC '07 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161219-151217993", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" }, { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Okawa Foundation" } ] }, "contributors": { "items": [ { "id": "Johnson-D", "name": { "family": "Johnson", "given": "David" } } ] }, "doi": "10.1145/1250790.1250854", "resource_type": "book_section", "pub_year": "2007", "author_list": "Shaltiel, Ronen and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/7ak5j-hfh67", "eprint_id": 76967, "eprint_status": "archive", "datestamp": "2023-08-19 20:23:54", "lastmod": "2023-10-25 17:02:35", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Guruswami-V", "name": { "family": "Guruswami", "given": "Venkatesan" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } }, { "id": "Vadhan-S", "name": { "family": "Vadhan", "given": "Salil" } } ] }, "title": "Unbalanced Expanders and Randomness Extractors from Parvaresh-Vardy Codes", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2007 IEEE. \n\nA preliminary version of this paper appeared on ECCC under the title \"Extractors and Condensers from Univariate Polynomials\" [10]. \n\nSupported by NSF CCF-0343672, a Sloan Research Fellowship, and a David and Lucile Packard Foundation Fellowship. \nSupported by NSF CCF-0346991, BSF 2004329, a Sloan Research Fellowship, and an Okawa Foundation research grant.\nSupported by NSF CCF-0133096, ONR N00014-04-1-0478, and US-Israel BSF 2002246.\n\nPublished - 04262755.pdf
", "abstract": "We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma, Umans, and Zuckerman (STOC '01) required at least one of these to be quasipolynomial in the optimal. \n\nOur expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy (FOCS \"05). Our expanders can be interpreted as near-optimal \"randomness condensers,\" that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. (STOC '3) and improving upon it when the error parameter is small (e.g. 1/poly(n)).", "date": "2007-06", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Piscataway, NJ", "pagerange": "96-108", "id_number": "CaltechAUTHORS:20170426-160240908", "isbn": "0-7695-2780-9", "book_title": "Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170426-160240908", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-0343672" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "David and Lucile Packard Foundation" }, { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" }, { "agency": "Okawa Foundation" }, { "agency": "NSF", "grant_number": "CCF-0133096" }, { "agency": "Office of Naval Research (ONR)", "grant_number": "N00014-04-1-0478" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2002246" } ] }, "doi": "10.1109/CCC.2007.38", "primary_object": { "basename": "04262755.pdf", "url": "https://authors.library.caltech.edu/records/7ak5j-hfh67/files/04262755.pdf" }, "resource_type": "book_section", "pub_year": "2007", "author_list": "Guruswami, Venkatesan; Umans, Christopher; et el." }, { "id": "https://authors.library.caltech.edu/records/smht2-rye23", "eprint_id": 98302, "eprint_status": "archive", "datestamp": "2023-08-22 07:43:45", "lastmod": "2024-01-14 21:55:17", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kalyanaraman-S", "name": { "family": "Kalyanaraman", "given": "Shankar" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "On Obtaining Pseudorandomness from Error-Correcting Codes", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Success Probability; Linear Code; Cyclic Code; Prediction Test; Linear Test", "note": "\u00a9 Springer-Verlag Berlin Heidelberg 2006.\n\nThis research was supported by NSF grant CCF-0346991 and by BSF grant 2004329.\n\nWe thank Eli Ben-Sasson for helpful discussions and Andrej Bogdanov\nfor sharing a draft of [Bog05] with us.We also thank the anonymous referees for\ntheir insightful comments.", "abstract": "A number of recent results have constructed randomness extractors and pseudorandom generators (PRGs) directly from certain error-correcting codes. The underlying construction in these results amounts to picking a random index into the codeword and outputting m consecutive symbols (the codeword is obtained from the weak random source in the case of extractors, and from a hard function in the case of PRGs). \n\nWe study this construction applied to general cyclic error-correcting codes, with the goal of understanding what pseudorandom objects it can produce. We show that every cyclic code with sufficient distance yields extractors that fool all linear tests. Further, we show that every polynomial code with sufficient distance yields extractors that fool all low-degree prediction tests. These are the first results that apply to univariate (rather than multivariate) polynomial codes, hinting that Reed-Solomon codes may yield good randomness extractors. \n\nOur proof technique gives rise to a systematic way of producing unconditional PRGs against restricted classes of tests. In particular, we obtain PRGs fooling all linear tests (which amounts to a construction of \u03b5-biased spaces), and we obtain PRGs fooling all low-degree prediction tests.", "date": "2006-12", "date_type": "published", "publisher": "Springer", "place_of_pub": "Berlin, Heidelberg", "pagerange": "105-116", "id_number": "CaltechAUTHORS:20190828-102318013", "isbn": "9783540499947", "book_title": "FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190828-102318013", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" } ] }, "contributors": { "items": [ { "id": "Arun-Kumar-S", "name": { "family": "Arun-Kumar", "given": "S." } }, { "id": "Garg-N", "name": { "family": "Garg", "given": "N." } } ] }, "doi": "10.1007/11944836_12", "resource_type": "book_section", "pub_year": "2006", "author_list": "Kalyanaraman, Shankar and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/3zvxb-z4977", "eprint_id": 77025, "eprint_status": "archive", "datestamp": "2023-08-19 18:42:04", "lastmod": "2023-10-25 17:14:14", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Ta-Shma-A", "name": { "family": "Ta-Shma", "given": "Amnon" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Better lossless condensers through derandomized curve samplers", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2006 IEEE.\n\nSupported by the Israel Science Foundation, by the Binational Science Foundation, and by the EU Integrated Project QAP. Supported by NSF Grant CCF-0346991, BSF Grant 2004329, and an Alfred P. Sloan Research Fellowship.\n\nPublished - 04031354.pdf
", "abstract": "Lossless condensers are unbalanced expander graphs, with expansion close to optimal. Equivalently, they may be viewed as functions that use a short random seed to map a source on n bits to a source on many fewer bits while preserving all of the min-entropy. It is known how to build lossless condensers when the graphs are slightly unbalanced in the work of M. Capalbo et al. (2002). The highly unbalanced case is also important but the only known construction does not condense the source well. We give explicit constructions of lossless condensers with condensing close to optimal, and using near-optimal seed length. Our main technical contribution is a randomness-efficient method for sampling FD (where F is a field) with low-degree curves. This problem was addressed before in the works of E. Ben-Sasson et al. (2003) and D. Moshkovitz and R. Raz (2006) but the solutions apply only to degree one curves, i.e., lines. Our technique is new and elegant. We use sub-sampling and obtain our curve samplers by composing a sequence of low-degree manifolds, starting with high-dimension, low-degree manifolds and proceeding through lower and lower dimension manifolds with (moderately) growing degrees, until we finish with dimension-one, low-degree manifolds, i.e., curves. The technique may be of independent interest.", "date": "2006-10", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Piscataway, NJ", "pagerange": "177-186", "id_number": "CaltechAUTHORS:20170427-151755097", "isbn": "0-7695-2720-5", "book_title": "47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170427-151755097", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Israel Science Foundation" }, { "agency": "European Union" }, { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "2004329" }, { "agency": "Alfred P. Sloan Foundation" } ] }, "doi": "10.1109/FOCS.2006.18", "primary_object": { "basename": "04031354.pdf", "url": "https://authors.library.caltech.edu/records/3zvxb-z4977/files/04031354.pdf" }, "resource_type": "book_section", "pub_year": "2006", "author_list": "Ta-Shma, Amnon and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/ettk9-9bw53", "eprint_id": 73192, "eprint_status": "archive", "datestamp": "2023-08-19 18:11:40", "lastmod": "2023-10-24 15:06:47", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Group-theoretic algorithms for matrix multiplication", "ispublished": "unpub", "full_text_status": "public", "keywords": "Algorithms, matrix multiplication, finite groups, representation theory", "note": "Copyright is held by the author/owner(s).\n\nPublished - p5-umans.pdf
", "abstract": "The exponent of matrix multiplication is the smallest real number \u03c9 such that for all \u03b5>0, O(n^(\u03c9+\u03b5)) arithmetic operations suffice to multiply two n\u00d7n matrices. The standard algorithm for matrix multiplication shows that \u03c9\u22643. Strassen's remarkable result [5] shows that \u03c9\u22642.81, and a sequence of further works culminating in the work of Coppersmith and Winograd [4] have improved this upper bound to \u03c9\u22642.376 (see [1] for a full history). Most researchers believe that in fact \u03c9=2, but there have been no further improvements in the known upper bounds for the past fifteen years. \n\nIt is known that several central linear algebra problems (for example, computing determinants, solving systems of equations, inverting matrices, computing LUP decompositions) have the same exponent as matrix multiplication, which makes \u03c9 a fundamental number for understanding algorithmic linear algebra. In addition, there are non-algebraic algorithms whose complexity is expressed in terms of \u03c9. \n\nIn this talk I will describe a new \"group-theoretic\" approach, proposed in [3], to devising algorithms for fast matrix multiplication. The basic idea is to reduce matrix multiplication to group algebra multiplication with respect to a suitable non-abelian group. The group algebra multiplication is performed in the Fourier domain, and then using this scheme recursively yields upper bounds on \u03c9. \n\nThis general framework produces nontrivial matrix multiplication algorithms if one can construct finite groups with certain properties. In particular, a very natural embedding of matrix multiplication into C[G]-multiplication is possible when group G has three subgroups H1, H2, H3 that satisfy the triple product property. I'll define this property and describe a construction that satisfies the triple product property with parameters that are necessary (but not yet sufficient) to achieve \u03c9=2. \n\nIn the next part of the talk I'll describe demands on the representation theory of the groups in order for the overall approach to yield non-trivial bounds on \u03c9, namely, that the character degrees must be \"small.\" Constructing families of groups together with subgroups satisfying the triple product property and for which the character degrees are sufficiently small has turned out to be quite challenging. \n\nIn [2], we succeed in constructing groups meeting both requirements, resulting in non-trivial algorithms for matrix multiplication in this framework. I'll outline the basic construction, together with more sophisticated variants that achieve the bounds \u03c9<2.48 and \u03c9<2.41. \n\nIn the final part of the talk I'll present two appealing conjectures, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.", "date": "2006-07", "date_type": "published", "publisher": "ACM", "place_of_pub": "New York, NY", "pagerange": "5", "id_number": "CaltechAUTHORS:20170103-170341500", "isbn": "1-59593-276-3", "book_title": "ISSAC '06 Proceedings of the 2006 international symposium on Symbolic and algebraic computation", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170103-170341500", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "contributors": { "items": [ { "id": "Trager-B", "name": { "family": "Trager", "given": "Barry" } } ] }, "doi": "10.1145/1145768.1145772", "primary_object": { "basename": "p5-umans.pdf", "url": "https://authors.library.caltech.edu/records/ettk9-9bw53/files/p5-umans.pdf" }, "resource_type": "book_section", "pub_year": "2006", "author_list": "Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/qvxkr-e0439", "eprint_id": 100950, "eprint_status": "archive", "datestamp": "2023-08-22 05:40:56", "lastmod": "2024-01-14 22:05:48", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Optimization Problems in the Polynomial-Time Hierarchy", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Approximation Algorithm; Boolean Circuit; SIGACT News; Constant Depth Circuit; Circuit Lower Bound", "note": "\u00a9 2006 Springer-Verlag Berlin Heidelberg. \n\nSupported by NSF grant CCF-0346991 and an Alfred P. Sloan Research Fellowship.", "abstract": "This talk surveys work on classifying the complexity and approximability of problems residing in the Polynomial-Time Hierarchy, above the first level. Along the way, we highlight some prominent natural problems that are believed \u2013 but not yet known \u2013 to be \u03a3^p\u2082-complete. We describe how strong inapproximability results for certain \u03a3^p\u2082 optimization problems can be obtained using dispersers to build error-correcting codes. Finally we adapt a learning algorithm to produce approximation algorithms for these problems.", "date": "2006-05-04", "date_type": "published", "publisher": "Springer", "place_of_pub": "Berlin", "pagerange": "345-355", "id_number": "CaltechAUTHORS:20200127-140148446", "isbn": "978-3-540-34021-8", "book_title": "Theory and Applications of Models of Computation", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200127-140148446", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCF-0346991" }, { "agency": "Alfred P. Sloan Foundation" } ] }, "contributors": { "items": [ { "id": "Cai-Jin-Yi", "name": { "family": "Cai", "given": "Jin-Yi" } }, { "id": "Cooper-S-B", "name": { "family": "Cooper", "given": "S. Barry" } }, { "id": "Li-Angsheng", "name": { "family": "Li", "given": "Angsheng" } } ] }, "doi": "10.1007/11750321_33", "resource_type": "book_section", "pub_year": "2006", "author_list": "Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/mm02b-r7m51", "eprint_id": 23966, "eprint_status": "archive", "datestamp": "2023-08-19 14:58:18", "lastmod": "2024-01-13 05:17:19", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Cohn-Henry", "name": { "family": "Cohn", "given": "Henry" } }, { "id": "Kleinberg-Robert", "name": { "family": "Kleinberg", "given": "Robert" } }, { "id": "Szegedy-Bal\u00e1zs", "name": { "family": "Szegedy", "given": "Bal\u00e1zs" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Group-theoretic Algorithms for Matrix Multiplication", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2005 IEEE. Date of Current Version: 14 November 2005. We are grateful to Michael Aschbacher, Noam Elkies, William Kantor, L\u00e1szl\u03cc Lov\u00e1sz, Amin Shokrollahi, G\u00e1bor\nSimonyi, and David Vogan for helpful discussions.\n\nPublished - COHfocs05.pdf
", "abstract": "We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.", "date": "2005", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Los Alamitos, CA", "pagerange": "379-388", "id_number": "CaltechAUTHORS:20110609-141427936", "isbn": "0-7695-2468-0", "book_title": "46th Annual IEEE Symposium on Foundations of Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110609-141427936", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "other_numbering_system": { "items": [ { "id": "8814628", "name": "INSPEC Accession Number" } ] }, "doi": "10.1109/SFCS.2005.39", "primary_object": { "basename": "COHfocs05.pdf", "url": "https://authors.library.caltech.edu/records/mm02b-r7m51/files/COHfocs05.pdf" }, "resource_type": "book_section", "pub_year": "2005", "author_list": "Cohn, Henry; Kleinberg, Robert; et el." }, { "id": "https://authors.library.caltech.edu/records/nc3n4-2n514", "eprint_id": 103795, "eprint_status": "archive", "datestamp": "2023-08-22 02:57:41", "lastmod": "2024-01-15 04:31:34", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Reconstructive Dispersers and Hitting Set Generators", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Pseudorandom Generator; 40th Annual IEEE Symposium; Oracle Access; Time Poly; Advice String", "note": "\u00a9 2005 Springer-Verlag Berlin Heidelberg. \n\nWe thank Ronen Shaltiel for his comments on an early draft of this paper.", "abstract": "We give a generic construction of an optimal hitting set generator (HSG) from any good \"reconstructive\" disperser. Past constructions of optimal HSGs have been based on such disperser constructions, but have had to modify the construction in a complicated way to meet the stringent efficiency requirements of HSGs. The construction in this paper uses existing disperser constructions with the \"easiest\" parameter setting in a black-box fashion to give new constructions of optimal HSGs without any additional complications. \n\nOur results show that a straightforward composition of the Nisan-Wigderson pseudorandom generator that is similar to the composition in works by Impagliazzo, Shaltiel and Wigderson in fact yields optimal HSGs (in contrast to the \"near-optimal\" HSGs constructed in those works). Our results also give optimal HSGs that do not use any form of hardness amplification or implicit list-decoding \u2013 like Trevisan's extractor, the only ingredients are combinatorial designs and any good list-decodable error-correcting code.", "date": "2005", "date_type": "published", "publisher": "Springer", "place_of_pub": "Berlin", "pagerange": "460-471", "id_number": "CaltechAUTHORS:20200609-101642741", "isbn": "978-3-540-28239-6", "book_title": "Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200609-101642741", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "contributors": { "items": [ { "id": "Chekuri-C", "name": { "family": "Chekuri", "given": "Chandra" } }, { "id": "Jansen-K", "name": { "family": "Jansen", "given": "Klaus" } }, { "id": "Rolim-J-D-P", "name": { "family": "Rolim", "given": "Jos\u00e9 D. P." } }, { "id": "Trevisan-L", "name": { "family": "Trevisan", "given": "Luca" } } ] }, "doi": "10.1007/11538462_39", "resource_type": "book_section", "pub_year": "2005", "author_list": "Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/vn3rk-z7618", "eprint_id": 27435, "eprint_status": "archive", "datestamp": "2023-08-19 12:04:51", "lastmod": "2023-10-24 17:09:14", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Cohn-H", "name": { "family": "Cohn", "given": "Henry" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "A Group-theoretic Approach to Fast Matrix Multiplication", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2003 IEEE. Date of Current Version: 20 October 2003. We are grateful to Michael Aschbacher, Noam Elkies, Bobby Kleinberg, L\u00e1szl\u03cc Lov\u00e1sz, Amin Shokrollahi, David Vogan, and Avi Wigderson for helpful discussions.", "abstract": "We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra C[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present machinery and examples to support (1), including a proof that certain families of groups of order n^(2+o(1)) support n \u00d7 n matrix multiplication, a necessary condition for the approach to yield exponent 2. Although we cannot yet completely achieve both (1) and (2), we hope that it may be possible, and we suggest potential routes to that result using the constructions in this paper.", "date": "2003-10", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Los Alamitos, CA", "pagerange": "438-449", "id_number": "CaltechAUTHORS:20111026-095124253", "isbn": "0-7695-2040-5", "book_title": "44th Annual IEEE Symposium on Foundations of Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111026-095124253", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "other_numbering_system": { "items": [ { "id": "7847069", "name": "INSPEC Accession Number" } ] }, "doi": "10.1109/SFCS.2003.1238217", "resource_type": "book_section", "pub_year": "2003", "author_list": "Cohn, Henry and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/snf03-ak987", "eprint_id": 27725, "eprint_status": "archive", "datestamp": "2023-08-19 09:23:17", "lastmod": "2024-01-13 05:46:48", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Pseudo-Random Generators for All Hardnesses", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2002 IEEE. Date of Current Version: 07 August 2002. The conference version of this abstract appears in the Proceedings of STOC 2002, May 19\u201321, 2002, Montreal, Quebec, Canada.", "abstract": "A pseudo-random generator (PRG) is a function that\n\"stretches\" a short random seed into a longer pseudo-random\noutput string that \"fools\" small circuits.", "date": "2002-05", "date_type": "published", "publisher": "IEEE", "place_of_pub": "Los Alamitos, CA", "pagerange": "7-7", "id_number": "CaltechAUTHORS:20111110-081820059", "isbn": "0-7695-1468-5", "book_title": "17th Annual Conference on Computational Complexity", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111110-081820059", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1109/CCC.2002.1004326", "resource_type": "book_section", "pub_year": "2002", "author_list": "Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/abqbr-ke227", "eprint_id": 27883, "eprint_status": "archive", "datestamp": "2023-08-19 08:19:31", "lastmod": "2024-01-13 05:48:17", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Shaltiel-R", "name": { "family": "Shaltiel", "given": "Ronen" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Simple Extractors for All Min-Entropies and a New Pseudo-Random Generator", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2001 IEEE. Date of Current Version: 07 August 2002. We thank Henry Cohn, Venkat Guruswami, Valentine Kabanets, Omer Reingold, Muli Safra, Amnon Ta-Shma, Salil Vadhan, Avi Wigderson and David Zuckerman for helpful discussions. We are especially grateful to the authors of [37] for explaining their result to us. We thank the conference referees for helpful comments.", "abstract": "We present a simple, self-contained extractor construction that produces good extractors for all min-entropies (min-entropy measures the amount of randomness contained in a weak random source). Our construction is algebraic and builds on a new polynomial-based approach introduced by A. Ta-Shma et al. (2001). Using our improvements, we obtain, for example, an extractor with output length m = k^(1-\u03b4) and seed length O(log n). This matches the parameters of L. Trevisan's (1999) breakthrough result and additionally achieves those parameters for small min-entropies k. Our construction gives a much simpler and more direct solution to this problem. Applying similar ideas to the problem of building pseudo-random generators, we obtain a new pseudo-random generator construction that is not based on the NW generator (N. Nisan and A. Widgerson, 1994), and turns worst-case hardness directly into pseudo-randomness. The parameters of this generator are strong enough to obtain a new proof that P=BPP if E requires exponential size circuits. Essentially, the same construction yields a hitting set generator with optimal seed length that outputs s^(\u03a9(1)) bits when given a function that requires circuits of size s (for any s). This implies a hardness versus randomness trade off for RP and BPP that is optimal (up to polynomial factors), solving an open problem raised by R. Impagliazzo et al. (1999). Our generators can also be used to derandomize AM in a way that improves\nand extends the results of [4, 18, 20].", "date": "2001-10", "date_type": "published", "publisher": "IEEE Computer Society", "place_of_pub": "Los Alamitos, CA", "pagerange": "648-657", "id_number": "CaltechAUTHORS:20111121-092803461", "isbn": "0-7695-1390-5", "book_title": "42nd Annual Symposium on Foundations of Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111121-092803461", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "other_numbering_system": { "items": [ { "id": "7121438", "name": "INSPEC Accession Number" } ] }, "doi": "10.1109/SFCS.2001.959941", "resource_type": "book_section", "pub_year": "2001", "author_list": "Shaltiel, Ronen and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/3gn2m-fc275", "eprint_id": 27860, "eprint_status": "archive", "datestamp": "2023-08-19 07:47:29", "lastmod": "2024-01-13 05:48:03", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Mossel-E", "name": { "family": "Mossel", "given": "Elchanan" }, "orcid": "0000-0001-7812-7886" }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "On the Complexity of Approximating the VC dimension", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 2001 IEEE. Date of Current Version: 07 August 2002. We thank Gil Kalai for helpful discussions and Adam Smith for a useful reference.", "abstract": "We study the complexity of approximating the VC dimension of a collection of sets, when the sets are encoded succinctly by a small circuit. We show that this problem is: \u03a3_3^p-hard to approximate to within a factor 2-\u03b5 for any \u03b5>0; approximable in AM to within a factor 2; and AM-hard to approximate to within a factor N_\u03b5 for some constant \u03b5>0. To obtain the \u03a3_3^p-hardness results we solve a randomness extraction problem using list-decodable binary codes; for the positive results we utilize the Sauer-Shelah(-Perles) Lemma. The exact value of \u03b5 in the AM-hardness result depends on the degree achievable by explicit disperser constructions.", "date": "2001-06", "date_type": "published", "publisher": "IEEE Computer Society", "place_of_pub": "Los Alamitos, CA", "pagerange": "220-225", "id_number": "CaltechAUTHORS:20111118-115638180", "isbn": "0-7695-1054-X", "book_title": "16th Annual IEEE Conference on Computational Complexity Proceedings", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111118-115638180", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "other_numbering_system": { "items": [ { "id": "6998461", "name": "INSPEC Accession Number" } ] }, "doi": "10.1109/CCC.2001.933889", "resource_type": "book_section", "pub_year": "2001", "author_list": "Mossel, Elchanan and Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/t01t1-md910", "eprint_id": 28718, "eprint_status": "archive", "datestamp": "2023-08-19 04:44:47", "lastmod": "2023-10-24 18:06:21", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "Hardness of Approximating \u03a3^(p)_(2) Minimization Problems", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 1999 IEEE.\n\nIssue Date: 1999; Date of Current Version: 06 August 2002.\nSupported in part by NSF grant CCR-9626361 and an NSF Graduate\nResearch Fellowship.\nWe wish to thank Christos Papadimitriou for many useful discussions.", "abstract": "We show that a number of natural optimization problems\nin the second level of the Polynomial Hierarchy are \u03a3^(p)_(2)-hard to approximate to within n factors, for specific \u03b5 > 0. The main technical tool is the use of explicit dispersers to achieve strong, direct inapproximability results. The problems we consider include Succinct Set Cover, Minimum Equivalent DNF, and other problems relating to DNF minimization. Under a slightly stronger complexity assumption, our method gives optimal n^(1-\u03b5) inapproximability results for some of these problems. We also prove inapproximability of a variant of an NP optimization problem, Monotone Minimum Satisfying Assignment, to within an n^\u03b5 factor using the same technique.", "date": "1999-10", "date_type": "published", "publisher": "IEEE Computer Society", "place_of_pub": "Los Alamitos, CA", "pagerange": "465-474", "id_number": "CaltechAUTHORS:20120109-142652598", "isbn": "0-7695-0409-4", "book_title": "40th Annual Symposium on Foundations of Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120109-142652598", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCR-9626361" }, { "agency": "NSF Graduate Research Fellowship" } ] }, "other_numbering_system": { "items": [ { "id": "6431136", "name": "INSPEC Accession Number" } ] }, "doi": "10.1109/SFFCS.1999.814619", "resource_type": "book_section", "pub_year": "1999", "author_list": "Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/081v8-b0580", "eprint_id": 28844, "eprint_status": "archive", "datestamp": "2023-08-19 03:28:01", "lastmod": "2024-01-13 05:50:42", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } } ] }, "title": "The Minimum Equivalent DNF Problem and Shortest Implicants", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 1998 IEEE. Issue Date: Nov 1998. Date of Current Version: 06 August 2002. Supported by NSF grant CCR-9626361 and an NSF Graduate Research Fellowship. We would like to thank Christos Papadimitriou for suggesting this problem, and many useful discussions.", "abstract": "We prove that the Minimum Equivalent DNF problem is _\u03a32 ^(p-)complete, resolving a conjecture due to L.J. Stockmeyer (1976). The proof involves as an intermediate step a variant of a related problem in logic minimization, namely, that of finding the shortest implicant of a Boolean function. We also obtain certain results concerning the complexity of the shortest implicant problem that may be of independent interest. When the input is a formula, the shortest implicant problem is \u03a3_2^(p-)complete, and \u03a3_2^(p-)hard to approximate to within an n^(1/2-\u03b5) factor. When the input is a circuit, approximation is \u03a3_2^(p-)hard to within an n^(1-\u03b5) factor. However, when the input is a DNF formula, the shortest implicant problem cannot be \u03a3_2^(p-)complete unless \u03a3_2^p=NP[log^2_n]^(NP).", "date": "1998-11", "date_type": "published", "publisher": "IEEE Computer Society", "place_of_pub": "Los Alamitos, CA", "pagerange": "556-563", "id_number": "CaltechAUTHORS:20120119-070934161", "isbn": "0-8186-9172-7", "book_title": "39th Annual Symposium on Foundations of Computer Science", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120119-070934161", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "CCR-9626361" }, { "agency": "NSF Graduate Research Fellowship" } ] }, "other_numbering_system": { "items": [ { "id": "6128822", "name": "INSPEC Accession Number" } ] }, "doi": "10.1109/SFCS.1998.743506", "resource_type": "book_section", "pub_year": "1998", "author_list": "Umans, Christopher" }, { "id": "https://authors.library.caltech.edu/records/ayywg-bkr43", "eprint_id": 28978, "eprint_status": "archive", "datestamp": "2023-08-19 01:51:39", "lastmod": "2023-10-24 18:17:09", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Umans-C", "name": { "family": "Umans", "given": "Christopher" } }, { "id": "Lenhart-W", "name": { "family": "Lenhart", "given": "William" } } ] }, "title": "Hamiltonian cycles in solid grid graphs", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 1997 IEEE. Date of Current Version: 06 August 2002.\nResearch supported in part by an NSF Graduate Research\nFellowship. We would like to thank Christos Papadimitriou for several useful comments on an earlier draft of this paper.", "abstract": "A grid graph is a finite node induced subgraph of the infinite two dimensional integer grid. A solid grid graph is a grid graph without holes. For general grid graphs, the Hamiltonian cycle problem is known to be NP complete. We give a polynomial time algorithm for the Hamiltonian cycle problem in solid grid graphs, resolving a longstanding open question posed by A. Itai et al. (1982). In fact, our algorithm can identify Hamiltonian cycles in quad quad graphs, a class of graphs that properly includes solid grid graphs.", "date": "1997-10", "date_type": "published", "publisher": "Los Alamitos, CA", "place_of_pub": "IEEE Computer Society Press", "pagerange": "496-505", "id_number": "CaltechAUTHORS:20120126-092849476", "isbn": "0-8186-8197-7", "book_title": "38th Annual Symposium on Foundations of Computer Science: Proceedings", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120126-092849476", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF Graduate Research Fellowship" } ] }, "other_numbering_system": { "items": [ { "id": "5809396", "name": "INSPEC Accession Number" } ] }, "doi": "10.1109/SFCS.1997.646138", "resource_type": "book_section", "pub_year": "1997", "author_list": "Umans, Christopher and Lenhart, William" } ]