[ { "id": "authors:byk5c-s6m73", "collection": "authors", "collection_id": "byk5c-s6m73", "cite_using_url": "https://authors.library.caltech.edu/records/byk5c-s6m73", "type": "article", "title": "Fast Generalized DFTs for All Finite Groups", "author": [ { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C" } ], "abstract": "

For any finite group G, we give an algebraic algorithm to compute the generalized discrete Fourier transform with respect to G, using O(|G|ω/2+\u03f5) operations, for any \u03f5>0. Here, ω is the exponent of matrix multiplication.

", "doi": "10.1137/20m1316342", "issn": "0097-5397", "publisher": "Society for Industrial & Applied Mathematics (SIAM)", "publication": "SIAM Journal on Computing", "publication_date": "2024-09-04", "pages": "FOCS19-398-FOCS19-419" }, { "id": "authors:mfnwn-hjx19", "collection": "authors", "collection_id": "mfnwn-hjx19", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191115-080137932", "type": "article", "title": "A New Algorithm for Fast Generalized DFTs", "author": [ { "family_name": "Hsu", "given_name": "Chloe Ching-Yun", "orcid": "0000-0002-7743-3168", "clpid": "Hsu-Chloe Ching-Yun" }, { "family_name": "Umans", "given_name": "Chris", "clpid": "Umans-C" } ], "abstract": "We give an new arithmetic algorithm to compute the generalized Discrete Fourier Transform (DFT) over finite groups G. The new algorithm uses O(\u2223G\u2223^(\u03c9 /2 + o(1))) operations to compute the generalized DFT over 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.\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\u2082(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.\nOur algorithm also yields an improved exponent for computing the generalized DFT over general finite groups G, which beats the longstanding previous best upper bound for any \u03c9. In particular, assuming \u03c9 = 2, we achieve exponent \u221a2, while the previous best was 3/2.", "doi": "10.1145/3301313", "issn": "1549-6325", "publisher": "Association for Computing Machinery (ACM)", "publication": "ACM Transactions on Algorithms", "publication_date": "2019-11", "series_number": "1", "volume": "16", "issue": "1", "pages": "Art. No. 4" }, { "id": "authors:wmv9a-40h47", "collection": "authors", "collection_id": "wmv9a-40h47", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191115-154547839", "type": "article", "title": "Algebraic Methods in Computational Complexity", "author": [ { "family_name": "Bl\u00e4ser", "given_name": "Markus", "clpid": "Bl\u00e4ser-M" }, { "family_name": "Kabanets", "given_name": "Valentine", "clpid": "Kabanets-V" }, { "family_name": "Tor\u00e1n", "given_name": "Jacobo", "clpid": "Tor\u00e1n-J" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test\nare some of the most prominent examples. In some of the most exciting recent progress in Computational Complexity the algebraic theme still plays a central role. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Also the areas of derandomization and coding theory have experimented important advances. The seminar aimed to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and the goal of the seminar was to play an important role in educating a diverse community about the latest new techniques.", "doi": "10.4230/DagRep.8.9.133", "issn": "2192-5283", "publisher": "Dagstuhl Publishing", "publication": "Dagstuhl Reports", "publication_date": "2018-09", "series_number": "9", "volume": "8", "issue": "9", "pages": "133-153" }, { "id": "authors:gqsk3-8rd15", "collection": "authors", "collection_id": "gqsk3-8rd15", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170531-152830870", "type": "article", "title": "On cap sets and the group-theoretic approach to matrix multiplication", "author": [ { "family_name": "Blasiak", "given_name": "Jonah", "clpid": "Blasiak-J" }, { "family_name": "Church", "given_name": "Thomas", "clpid": "Church-T" }, { "family_name": "Cohn", "given_name": "Henry", "clpid": "Cohn-H" }, { "family_name": "Grochow", "given_name": "Joshua A.", "clpid": "Grochow-J-A" }, { "family_name": "Naslund", "given_name": "Eric", "clpid": "Naslund-E" }, { "family_name": "Sawin", "given_name": "William F.", "clpid": "Sawin-W-F" }, { "family_name": "Umans", "given_name": "Chris", "clpid": "Umans-C" } ], "abstract": "In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent \u03c9 of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain \u03c9 = 2. In this paper we rule out obtaining \u03c9 = 2 in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory.", "doi": "10.19086/da.1245", "issn": "2397-3129", "publisher": "Discrete Analysis", "publication": "Discrete Analysis", "publication_date": "2017-01-16", "series_number": "3", "volume": "2017", "issue": "3", "pages": "1-27" }, { "id": "authors:s7nkn-rq096", "collection": "authors", "collection_id": "s7nkn-rq096", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191118-153821359", "type": "article", "title": "Algebraic and Combinatorial Methods in Computational Complexity", "author": [ { "family_name": "Kabanets", "given_name": "Valentine", "clpid": "Kabanets-V" }, { "family_name": "Thierauf", "given_name": "Thomas", "clpid": "Thierauf-T" }, { "family_name": "Tor\u00e1n", "given_name": "Jacobo", "clpid": "Tor\u00e1n-J" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example, Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms. This emphases once again the central role of algebra in computer science. The seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar can play an important role in educating a diverse community about the latest new techniques, spurring further progress.", "doi": "10.4230/DagRep.6.10.13", "issn": "2192-5283", "publisher": "Dagstuhl Publishing", "publication": "Dagstuhl Reports", "publication_date": "2016-10", "series_number": "10", "volume": "6", "issue": "10", "pages": "13-32" }, { "id": "authors:zra1e-qjk46", "collection": "authors", "collection_id": "zra1e-qjk46", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-133603851", "type": "article", "title": "Algebra in Computational Complexity", "author": [ { "family_name": "Agrawal", "given_name": "Manindra", "clpid": "Agrawal-M" }, { "family_name": "Kabanets", "given_name": "Valentine", "clpid": "Kabanets-V" }, { "family_name": "Thierauf", "given_name": "Thomas", "clpid": "Thierauf-T" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called \"chasm at depth 4\" suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model, and these are tied to central questions regarding the power of randomness in computation. Representation theory has emerged as an important tool in three separate lines of work: the \"Geometric Complexity Theory\" approach to P vs. NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes. Coding theory has seen several algebraic innovations in recent years, including multiplicity codes, and new lower bounds. This seminar brought together researchers who are using a diverse array of algebraic methods in a variety of settings. It plays an important role in educating a diverse community about the latest new techniques, spurring further progress.", "doi": "10.4230/DagRep.4.9.85", "issn": "2192-5283", "publisher": "Dagstuhl Publishing", "publication": "Dagstuhl Reports", "publication_date": "2014-09", "series_number": "9", "volume": "4", "issue": "9", "pages": "85-105" }, { "id": "authors:v0nha-rpf57", "collection": "authors", "collection_id": "v0nha-rpf57", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140701-090549753", "type": "article", "title": "Special Issue \"Conference on Computational Complexity 2013\" Guest editor's foreword", "author": [ { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "This special issue contains the full versions of five papers that were\npresented at the 28th Annual IEEE Conference on Computational\nComplexity (CCC 2013) held in Palo Alto, California, from June\n5, 2013 to June 7, 2013. These outstanding papers were selected in\nconsultation with the program committee from among the twentynine\npapers that appeared in the conference. They were invited for\nsubmission and subsequently subjected to the standard refereeing\nprocess of the journal.", "doi": "10.1007/s00037-014-0088-x", "issn": "1016-3328", "publisher": "Springer", "publication": "Computational Complexity", "publication_date": "2014-06", "series_number": "2", "volume": "23", "issue": "2", "pages": "147-149" }, { "id": "authors:8kb7d-trj22", "collection": "authors", "collection_id": "8kb7d-trj22", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191118-154821951", "type": "article", "title": "On Beating the Hybrid Argument", "author": [ { "family_name": "Fefferman", "given_name": "Bill", "clpid": "Fefferman-B" }, { "family_name": "Shaltiel", "given_name": "Ronen", "clpid": "Shaltiel-R" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" }, { "family_name": "Viola", "given_name": "Emanuele", "clpid": "Viola-E" } ], "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: \u03f5-distinguishability implies \u03f5/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. Our main results are:\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\u2070. We conjecture that this generator indeed fools AC\u2070. 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. (This was also observed by other researchers.) 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\nG^(\u2297k)_f(x\u2081,\u2026,x_k)=(x\u2081,f(x\u2081),x\u2082,f(x\u2082),\u2026,x_k,f(x_k)) \nfrom uniform. This gives new pseudorandom generators for classes such as AC\u2070 [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^(\u2297k)_f on correlated x\u2081,\u2026,x_k) and proving the conjecture in the first item.\nAn extended abstract of this paper appeared in the Proceedings of the 3rd Innovations in Theoretical Computer Science (ITCS) 2012.", "doi": "10.4086/toc.2013.v009a026", "issn": "1557-2862", "publisher": "University of Chicago", "publication": "Theory of Computing", "publication_date": "2013-11-14", "series_number": "1", "volume": "9", "issue": "1", "pages": "809-843" }, { "id": "authors:pm0dy-mrb85", "collection": "authors", "collection_id": "pm0dy-mrb85", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130703-102109533", "type": "article", "title": "On sunflowers and matrix multiplication", "author": [ { "family_name": "Alon", "given_name": "Noga", "orcid": "0000-0003-1332-4883", "clpid": "Alon-Noga" }, { "family_name": "Shpilka", "given_name": "Amir", "clpid": "Shpilka-A" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "We present several variants of the sunflower conjecture of Erd\u0151s & Rado (J Lond Math Soc 35:85\u201390, 1960) and discuss the relations among them.\nWe then show that two of these conjectures (if true) imply negative answers to the questions of Coppersmith & Winograd (J Symb Comput 9:251\u2013280, 1990) and Cohn et al. (2005) regarding possible approaches for obtaining fast matrix-multiplication algorithms. Specifically, we show that the Erd\u0151s\u2013Rado sunflower conjecture (if true) implies a negative answer to the \"no three disjoint equivoluminous subsets\" question of Coppersmith & Winograd (J Symb Comput 9:251\u2013280, 1990); 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 Cohn et al. (2005) (although it does not seem to impact a second conjecture in Cohn et al. (2005) or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith\u2013Winograd conjecture actually implies the Cohn et al. conjecture.\nThe 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 Cohn et al. (2005) 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 Edel (2004) on the size of the largest 3-sunflower-free set in Z^n_3.", "doi": "10.1007/s00037-013-0060-1", "issn": "1016-3328", "publisher": "Springer", "publication": "Computational Complexity", "publication_date": "2013-06", "series_number": "2", "volume": "22", "issue": "2", "pages": "219-243" }, { "id": "authors:2719d-ve992", "collection": "authors", "collection_id": "2719d-ve992", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-134942908", "type": "article", "title": "Special Section on the Forty-First Annual ACM Symposium on Theory of Computing (STOC 2009)", "author": [ { "family_name": "Immorlica", "given_name": "Nicole", "clpid": "Immorlica-N" }, { "family_name": "Katz", "given_name": "Jonathan N.", "orcid": "0000-0002-5287-3503", "clpid": "Katz-J-N" }, { "family_name": "Mitzenmacher", "given_name": "Michael", "clpid": "Mitzenmacher-M" }, { "family_name": "Servedio", "given_name": "Rocco", "clpid": "Servedio-R" }, { "family_name": "Umans", "given_name": "Chris", "clpid": "Umans-C" } ], "abstract": "This issue of SICOMP contains nine specially selected papers from the Forty-first Annual ACM Symposium on the Theory of Computing, otherwise known as STOC 2009, held May 31 to June 2 in Bethesda, Maryland. The papers here were chosen to represent both the excellence and the broad range of the STOC program. The papers have been revised and extended by the authors, and subjected to the standard thorough reviewing process of SICOMP.\n\nThe program committee consisted of Susanne Albers, Andris Ambainis, Nikhil Bansal, Paul Beame, Andrej Bogdanov, Ran Canetti, David Eppstein, Dmitry Gavinsky, Shafi Goldwasser, Nicole Immorlica, Anna Karlin, Jonathan Katz, Jonathan Kelner, Subhash Khot, Ravi Kumar, Leslie Ann Goldberg, Michael Mitzenmacher (Chair), Kamesh Munagala, Rasmus Pagh, Anup Rao, Rocco Servedio, Mikkel Thorup, Chris Umans, and Lisa Zhang. They accepted 77 papers out of 321 submissions.", "doi": "10.1137/120973305", "issn": "0097-5397", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Journal on Computing", "publication_date": "2012-12-18", "series_number": "6", "volume": "41", "issue": "6", "pages": "1591-1592" }, { "id": "authors:e8rpy-ybr95", "collection": "authors", "collection_id": "e8rpy-ybr95", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-143839781", "type": "article", "title": "Algebraic and Combinatorial Methods in Computational\n Complexity", "author": [ { "family_name": "Agrawal", "given_name": "Manindra", "clpid": "Agrawal-M" }, { "family_name": "Thierauf", "given_name": "Thomas", "clpid": "Thierauf-T" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings.", "doi": "10.4230/DagRep.2.10.60", "issn": "2192-5283", "publisher": "Dagstuhl Publishing", "publication": "Dagstuhl Reports", "publication_date": "2012-10", "series_number": "10", "volume": "2", "issue": "10", "pages": "60-78" }, { "id": "authors:m9qjm-tht77", "collection": "authors", "collection_id": "m9qjm-tht77", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120125-151548395", "type": "article", "title": "Fast Polynomial Factorization and Modular Composition", "author": [ { "family_name": "Kedlaya", "given_name": "Kiran S.", "clpid": "Kedlaya-K-S" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "We obtain randomized algorithms for factoring degree n univariate polynomials over F_q requiring O(n^(1.5+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 faster than the best previous algorithms [J. von zur Gathen and V. Shoup, Comput.\nComplexity, 2 (1992), pp. 187\u2013224; E. Kaltofen and V. Shoup, Math. Comp., 67 (1998), pp. 1179\u20131197]; for log q \u2265 n, it matches the asymptotic running time of the best known algorithms. The 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. The best previous algorithms for modular composition use O(n^((\u03c9+1)/2)) field operations, where \u03c9 is the exponent of matrix multiplication [R. P. Brent and H. T. Kung, J. Assoc. Comput. Mach., 25 (1978), pp. 581\u2013595], with a slight improvement in the exponent achieved by employing fast\nrectangular matrix multiplication [X. Huang and V. Y. Pan, J. Complexity, 14 (1998), pp. 257\u2013299]. We 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 two new algorithms that\nsolve the problem near-optimally: an algebraic algorithm for fields of characteristic at most n^(o(1)), and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of multimodular reduction, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese remainder theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithms use techniques that are commonly employed in practice, in contrast to all previous subquadratic algorithms for these problems, which relied on fast matrix multiplication.", "doi": "10.1137/08073408X", "issn": "0097-5397", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Journal on Computing", "publication_date": "2011-12-22", "series_number": "6", "volume": "40", "issue": "6", "pages": "1767-1802" }, { "id": "authors:epgka-nze16", "collection": "authors", "collection_id": "epgka-nze16", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-142038137", "type": "article", "title": "On Sunflowers and Matrix Multiplication", "author": [ { "family_name": "Alon", "given_name": "Noga", "orcid": "0000-0003-1332-4883", "clpid": "Alon-Noga" }, { "family_name": "Shpilka", "given_name": "Amir", "clpid": "Shpilka-A" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "We present several variants of the sunflower conjecture of Erd\u0151s and Rado and discuss the relations among them.\nWe then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd and Cohn et al. regarding possible approaches for obtaining fast matrix multiplication algorithms. Specifically, we show that the Erd\u0151s-Rado sunflower conjecture (if true) implies a negative answer to the \"no three disjoint equivoluminous subsets\" question of Coppersmith and Winograd; we also formulate a \"multicolored\" sunflower conjecture in Zn\u2083 and show that (if true) it implies a negative answer to the \"strong USP\" conjecture of Cohn et al. (although it does not seem to impact a second conjecture in that paper or the viability of the general group theoretic approach). A surprising consequence of our results is that the Coppersmith-Winograd conjecture actually implies the Cohn et al. conjecture.\nThe multicolored sunflower conjecture in Zn\u2083 is a strengthening of the well-known (ordinary) sunflower conjecture in Zn\u2083, and we show via our connection that a construction of Cohn et al. 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 on the size of the largest 3-sunflower-free set in Zn\u2083.", "issn": "1433-8092", "publisher": "Computational Complexity Foundation (CCF)", "publication": "Electronic Colloquium on Computational Complexity", "publication_date": "2011-04-25", "volume": "2011", "pages": "Art. No. 67" }, { "id": "authors:t2v4w-6ps74", "collection": "authors", "collection_id": "t2v4w-6ps74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110422-100438828", "type": "article", "title": "The complexity of Boolean formula minimization", "author": [ { "family_name": "Buchfuhrer", "given_name": "David", "clpid": "Buchfuhrer-D" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "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_2-complete and indeed appears as an open problem in Garey and Johnson (1979) [5]. The depth-2 variant was only shown to be \u03a3^P_2-complete in 1998 (Umans (1998) [13], Umans (2001) [15]) 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_2-complete under Turing reductions for all k \u2265 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is \u03a3^P_2-complete under Turing reductions.", "doi": "10.1016/j.jcss.2010.06.011", "issn": "0022-0000", "publisher": "Elsevier", "publication": "Journal of Computer and System Sciences", "publication_date": "2011-01", "series_number": "1", "volume": "77", "issue": "1", "pages": "142-153" }, { "id": "authors:af5ze-8d022", "collection": "authors", "collection_id": "af5ze-8d022", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-153323226", "type": "article", "title": "On beating the hybrid argument", "author": [ { "family_name": "Fefferman", "given_name": "Bill", "clpid": "Fefferman-B" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" }, { "family_name": "Shaltiel", "given_name": "Ronen", "clpid": "Shaltiel-R" }, { "family_name": "Viola", "given_name": "Emanuele", "clpid": "Viola-E" } ], "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\nk equal to the bit-length of the distributions: \u03f5-distinguishability implies only \u03f5/k-predictability.\nThis 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.\nSpecifically, we obtain the following results:\n1. We give an instantiation of the Nisan-Wigderson generator (JCSS '94) that can be broken\nby quantum computers, and that is o(1)-unpredictable against AC\u2070. This is not enough\nto imply indistinguishability via the hybrid argument because of the hybrid-argument\nloss; nevertheless, we conjecture that this generator indeed fools AC\u2070, and we prove this\nstatement for a simplified version of the problem. Our conjecture implies the existence of\nan 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)\nwith seed length O(log n log log n) produces a distribution that is 1= log n-unpredictable\nagainst poly-logarithmic width (general) read-once oblivious branching programs. Thus\navoiding the hybrid-argument loss would lead to a breakthrough in generators against\nsmall space.\n3. We study pseudorandom generators obtained from a hard function by repeated sampling.\nWe identify a property of functions, \"resamplability,\" that allows us to beat the hybrid argument, leading to new pseudorandom generators for AC\u2070[p] and similar classes. Although\nthe generators have sub-linear stretch, they represent the best-known generators for these\nclasses.\nThus we establish that \"beating\" or bypassing the hybrid argument would have two significant consequences in complexity, and we take steps toward that goal by developing techniques that indeed beat the hybrid argument in related (but simpler) settings, leading to best-known PRGs for certain complexity classes.", "issn": "1433-8092", "publisher": "Computational Complexity Foundation (CCF)", "publication": "Electronic Colloquium on Computational Complexity", "publication_date": "2010-12-02", "volume": "2010", "pages": "Art. No. 186" }, { "id": "authors:vg9e2-dnz47", "collection": "authors", "collection_id": "vg9e2-dnz47", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191126-150500984", "type": "article", "title": "Special Section On Foundations of Computer Science", "author": [ { "family_name": "Lee", "given_name": "James R.", "clpid": "Lee-James-R" }, { "family_name": "Umans", "given_name": "Chris", "clpid": "Umans-C" } ], "abstract": "This special section comprises eight fully refereed papers whose extended abstracts were presented at the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007) in Providence, Rhode Island, October 21\u201323, 2007. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2007 proceedings.", "doi": "10.1137/smjcat000039000006002397000001", "issn": "0097-5397", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Journal on Computing", "publication_date": "2010-04-30", "series_number": "6", "volume": "39", "issue": "6", "pages": "2397-2397" }, { "id": "authors:28ynw-3cm55", "collection": "authors", "collection_id": "28ynw-3cm55", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191127-082201341", "type": "article", "title": "The Complexity of Rationalizing Network Formation", "author": [ { "family_name": "Kalyanaraman", "given_name": "Shankar", "clpid": "Kalyanaraman-S" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "We study the complexity of rationalizing network formation. In this problem we fix an underlying\nmodel describing how selfish parties (the vertices) produce a graph by making individual decisions to\nform or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and\nwe observe a set of \"snapshots\" of graphs that are assumed to be stable. From this we would like to infer\nsome unobserved data about the system: edge prices, or how much each vertex values short paths to each\nother vertex.\nWe study two rationalization problems arising from the network formation model of Jackson and\nWolinsky [JW96]. When the goal is to infer edge prices, we observe that the rationalization problem is\neasy. The problem remains easy even when rationalizing prices do not exist and we instead wish to find\nprices that maximize the stability of the system.\nIn contrast, when the edge prices are given and the goal is instead to infer valuations of each vertex\nby each other vertex, we prove that the rationalization problem becomes NP-hard. Our proof exposes a\nclose connection between rationalization problems and the Inequality-SAT (I-SAT) problem.\nFinally and most significantly, we prove that an approximation version of this NP-complete rationalization\nproblem is NP-hard to approximate to within better than a 1/2 ratio. This shows that the trivial\nalgorithm of setting everyone's valuations to infinity (which rationalizes all the edges present in the input\ngraphs) or to zero (which rationalizes all the non-edges present in the input graphs) is the best possible\nassuming P \u2260 NP. To do this we prove a tight (1/2+\u03b4)-approximation hardness for a variant of I-SAT in\nwhich all coefficients are non-negative. This in turn follows from a tight hardness result for MAX-LIN_(R+)\n(linear equations over the reals, with non-negative coefficients), which we prove by a (non-trivial) modification\nof the recent result of Guruswami and Raghavendra [GR07] which achieved tight hardness for\nthis problem without the non-negativity constraint.\nOur technical contributions regarding the hardness of I-SAT and MAX-LIN_(R+) may be of independent\ninterest, given the generality of these problems.", "issn": "1433-8092", "publisher": "Computational Complexity Foundation (CCF)", "publication": "Electronic Colloquium on Computational Complexity", "publication_date": "2009-12-19", "volume": "2009", "pages": "Art. No. 145" }, { "id": "authors:fhv5w-h3x81", "collection": "authors", "collection_id": "fhv5w-h3x81", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191127-080702908", "type": "article", "title": "Improved inapproximability factors for some \u03a3^p\u2082 minimization problems", "author": [ { "family_name": "Dick", "given_name": "Kevin", "clpid": "Dick-K" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "We give improved inapproximability results for some minimization problems in the second level of the Polynomial-Time Hierarchy. Extending previous work by Umans [Uma99], we show that several variants of DNF minimization are \u03a3^p\u2082-hard to approximate to within factors of n^(1/3\u2212\u03f5) and ^(n1/2\u2212\u03f5) (where the previous results achieved n^(1/4\u2212\u03f5)), for arbitrarily small constant \u03f5 > 0. For one problem shown to be inapproximable to within n^(1/2\u2212\u03f5), we give a matching O(n^(1/2))-approximation algorithm, running in randomized polynomial time with access to an NP oracle, which shows that this result is tight assuming the PH doesn't collapse.", "issn": "1433-8092", "publisher": "Computational Complexity Foundation (CCF)", "publication": "Electronic Colloquium on Computational Complexity", "publication_date": "2009-10-28", "volume": "2009", "pages": "Art. No. 107" }, { "id": "authors:fzepj-9xh02", "collection": "authors", "collection_id": "fzepj-9xh02", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20091023-111539349", "type": "article", "title": "Low-End Uniform Hardness versus Randomness Tradeoffs for AM", "author": [ { "family_name": "Shaltiel", "given_name": "Ronen", "clpid": "Shaltiel-R" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "Impagliazzo and Wigderson [Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Washington, DC, 1998, pp. 734\u2013743]\nproved a hardness versus randomness tradeoff for BPP in the uniform setting, which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions (in slightly weaker settings). Gutfreund, Shaltiel, and Ta-Shma [Comput. Complexity, 12 (2003), pp. 85\u2013130] proved a uniform hardness versus randomness tradeoff for AM, but that result worked only on the\n\"high end\" of possible hardness assumptions. In this work, we give uniform hardness versus randomness tradeoffs for AM that are near-optimal for the full range of possible hardness assumptions. Following Gutfreund, Shaltiel, and Ta-Shma, we do this by constructing a hitting-set-generator (HSG) for AM with \"resilient reconstruction.\" Our construction is a recursive variant of the Miltersen\u2013\nVinodchandran HSG [Comput. Complexity, 14 (2005), pp. 256\u2013279], 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\u2013Muller codes for a\nsort of locally computable error-reduction. As 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\u2013Merlin games are powerful\") or else all of AM (or AM \u2229 coAM) can be simulated in nondeterministic time s(n) (\"Arthur\u2013Merlin games can be derandomized\") for\na near-optimal relationship between t(n) and s(n). As in Gutfreund, Shatiel, and Ta-Shma, 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.", "doi": "10.1137/070698348", "issn": "0097-5397", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Journal on Computing", "publication_date": "2009-09-02", "series_number": "3", "volume": "39", "issue": "3", "pages": "1006-1037" }, { "id": "authors:mw6pc-7dd24", "collection": "authors", "collection_id": "mw6pc-7dd24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090901-145722580", "type": "article", "title": "Reconstructive Dispersers and Hitting Set Generators", "author": [ { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "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. \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\u2014like Trevisan's extractor, the only ingredients are combinatorial designs and any good list-decodable error-correcting code.", "doi": "10.1007/s00453-008-9266-z", "issn": "0178-4617", "publisher": "Springer", "publication": "Algorithmica", "publication_date": "2009-09", "series_number": "1", "volume": "55", "issue": "1", "pages": "134-156" }, { "id": "authors:ydt3m-73k86", "collection": "authors", "collection_id": "ydt3m-73k86", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191127-082848396", "type": "article", "title": "Limits on the Social Welfare of Maximal-In-Range Auction Mechanisms", "author": [ { "family_name": "Buchfuhrer", "given_name": "David", "clpid": "Buchfuhrer-D" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "Many commonly-used auction mechanisms are \"maximal-in-range\". We show that any maximal-in-range mechanism for n bidders and m items cannot both approximate the social welfare with a ratio better than min(n m^\u03b7) for any constant \u03b7 < 1/2 and run in polynomial time, unless NP \u2286 P poly. This significantly improves upon a previous bound on the achievable social welfare of polynomial time maximal-in-range mechanisms of 2n/(n+1) for constant n. Our bound is tight, as a min(n, 2m^(1/2) approximation of the social welfare is achievable.", "issn": "1433-8092", "publisher": "Computational Complexity Foundation (CCF)", "publication": "Electronic Colloquium on Computational Complexity", "publication_date": "2009-08-06", "volume": "2009", "pages": "Art. No. 68" }, { "id": "authors:jy06s-8qb83", "collection": "authors", "collection_id": "jy06s-8qb83", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090817-144815953", "type": "article", "title": "Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes", "author": [ { "family_name": "Guruswami", "given_name": "Venkatesan", "clpid": "Guruswami-V" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" }, { "family_name": "Vadhan", "given_name": "Salil", "clpid": "Vadhan-S" } ], "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 et al. [2007] required at least one of these to be quasipolynomial in the optimal. Our 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 [2005]. \n\nOur 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, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. [2003] and improving upon it when the error parameter is small (e.g., 1/poly(n)).", "doi": "10.1145/1538902.1538904", "issn": "0004-5411", "publisher": "Association for Computing Machinery", "publication": "Journal of the ACM", "publication_date": "2009-06", "series_number": "4", "volume": "56", "issue": "4", "pages": "20" }, { "id": "authors:n1c9f-krc68", "collection": "authors", "collection_id": "n1c9f-krc68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090422-111344642", "type": "article", "title": "The complexity of the matroid-greedoid partition problem", "author": [ { "family_name": "Asodi", "given_name": "Vera", "clpid": "Asodi-Vera" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "We show that the maximum matroid\u2013greedoid partition problem is NP-hard to approximate to within 1/2+\u03b5 for any \u03b5>0, which matches the trivial factor 1/2 approximation algorithm. The main tool in our hardness of approximation result is an extractor code with polynomial rate, alphabet size and list size, together with an efficient algorithm for list-decoding. We show that the recent extractor construction of Guruswami, Umans and Vadhan [V. Guruswami, C. Umans, S.P. Vadhan, Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes, in: IEEE Conference on Computational Complexity, IEEE Computer Society, 2007, pp. 96\u2013108] can be used to obtain a code with these properties. We also show that the parameterized matroid\u2013greedoid partition problem is fixed-parameter tractable.", "doi": "10.1016/j.tcs.2008.11.019", "issn": "0304-3975", "publisher": "Elsevier", "publication": "Theoretical Computer Science", "publication_date": "2009-03-01", "series_number": "8-10", "volume": "410", "issue": "8-10", "pages": "859-866" }, { "id": "authors:j8nbh-rcd32", "collection": "authors", "collection_id": "j8nbh-rcd32", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:FORcc08", "type": "article", "title": "On the Complexity of Succinct Zero-Sum Games", "author": [ { "family_name": "Fortnow", "given_name": "Lance", "clpid": "Fortnow-L" }, { "family_name": "Impagliazzo", "given_name": "Russell", "clpid": "Impagliazzo-R" }, { "family_name": "Kabanets", "given_name": "Valentine", "clpid": "Kabanets-V" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i,j) = C(i,j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approximating the value of a succinct zero-sum game to within an additive error is complete for the class promise-$$S^{p}_{2}$$, the \"promise\" version of $$S^{p}_{2}$$. To the best of our knowledge, it is the first natural problem shown complete for this class. (2) We describe a ZPP NP algorithm for constructing approximately optimal strategies, and hence for approximating the value, of a given succinct zero-sum game. As a corollary, we obtain, in a uniform fashion, several complexity-theoretic results, e.g., a ZPP NP algorithm for learning circuits for SAT (Bshouty et al., JCSS, 1996) and a recent result by Cai (JCSS, 2007) that $$S^{p}_{2} \u2286 ZPP NP. (3) We observe that approximating the value of a succinct zero-sum game to within a multiplicative factor is in PSPACE, and that it cannot be in promise-$$S^{p}_{2}$$ unless the polynomial-time hierarchy collapses. Thus, under a reasonable complexity-theoretic assumption, multiplicative-factor approximation of succinct zero-sum games is strictly harder than additive-error approximation.", "doi": "10.1007/s00037-008-0252-2", "issn": "1016-3328", "publisher": "Springer", "publication": "Computational Complexity", "publication_date": "2008-10", "series_number": "3", "volume": "17", "issue": "3", "pages": "353-376" }, { "id": "authors:hj5nx-af624", "collection": "authors", "collection_id": "hj5nx-af624", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191127-084809601", "type": "article", "title": "The Complexity of Rationalizing Matchings", "author": [ { "family_name": "Kalyanaraman", "given_name": "Shankar", "clpid": "Kalyanaraman-S" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "Given a set of observed economic choices, can one infer preferences and/or utility functions for the\nplayers that are consistent with the data? Questions of this type are called rationalization or revealed\npreference problems in the economic literature, and are the subject of a rich body of work.\nFrom the computer science perspective, it is natural to study the complexity of rationalization in various\nscenarios. We consider a class of rationalization problems in which the economic data is expressed\nby a collection of matchings, and the question is whether there exist preference orderings for the nodes\nunder which all the matchings are stable.\nWe show that the rationalization problem for one-one matchings is NP-complete. We propose two\nnatural notions of approximation, and show that the problem is hard to approximate to within a constant\nfactor, under both. On the positive side, we describe a simple algorithm that achieves a 3/4 approximation\nratio for one of these approximation notions. We also prove similar results for a version of many-one matching.", "issn": "1433-8092", "publisher": "Computational Complexity Foundation (CCF)", "publication": "Electronic Colloquium on Computational Complexity", "publication_date": "2008-03-02", "pages": "Art. No. 21" }, { "id": "authors:z9v21-my692", "collection": "authors", "collection_id": "z9v21-my692", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110811-085055181", "type": "article", "title": "Pseudorandomness for Approximate Counting and Sampling", "author": [ { "family_name": "Shaltiel", "given_name": "Ronen", "clpid": "Shaltiel-R" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions.\nOur main technical contribution allows one to \"boost\" a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent.\nWe also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the \"boosting\" theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM.\nWe observe that Cai's proof that S_2^P \u2286 PP\u2286(NP) and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice.", "doi": "10.1007/s00037-007-0218-9", "issn": "1016-3328", "publisher": "Springer", "publication": "Computational Complexity", "publication_date": "2006-12", "series_number": "4", "volume": "15", "issue": "4", "pages": "298-341" }, { "id": "authors:0rzht-sf015", "collection": "authors", "collection_id": "0rzht-sf015", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190829-152850467", "type": "article", "title": "On obtaining pseudorandomness from error-correcting codes", "author": [ { "family_name": "Kalyanaraman", "given_name": "Shankar", "clpid": "Kalyanaraman-S" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "A number of recent results have constructed randomness extractors\nand pseudorandom generators (PRGs) directly from certain\nerror-correcting codes. The underlying construction in these\nresults amounts to picking a random index into the codeword and\noutputting m consecutive symbols (the codeword is obtained from\nthe weak random source in the case of extractors, and from a hard\nfunction in the case of PRGs).\n\nWe study this construction applied to general cyclic\nerror-correcting codes, with the goal of understanding what\npseudorandom objects it can produce. We show that every cyclic code with sufficient distance yields extractors that fool\nall linear tests. Further, we show that every polynomial\ncode with sufficient distance yields extractors that fool all\nlow-degree prediction tests. These are the first results that\napply to univariate (rather than multivariate) polynomial codes,\nhinting that Reed-Solomon codes may yield good randomness\nextractors.\n\nOur proof technique gives rise to a systematic way of producing\nunconditional PRGs against restricted classes of tests. In\nparticular, we obtain PRGs fooling all linear tests (which amounts\nto a construction of epsilon-biased spaces), and we obtain PRGs\nfooling all low-degree prediction tests.", "issn": "1433-8092", "publisher": "Computational Complexity Foundation (CCF)", "publication": "Electronic Colloquium on Computational Complexity", "publication_date": "2006-10-09", "volume": "2006", "pages": "TR06-128" }, { "id": "authors:pwqs2-7z819", "collection": "authors", "collection_id": "pwqs2-7z819", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:UMAieeetcadics06", "type": "article", "title": "Complexity of two-level logic minimization", "author": [ { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" }, { "family_name": "Villa", "given_name": "Tiziano", "clpid": "Villa-T" }, { "family_name": "Sangiovanni-Vincentelli", "given_name": "Alberto L.", "orcid": "0000-0003-1298-8389", "clpid": "Sangiovanni-Vincentelli-A-L" } ], "abstract": "The complexity of two-level logic minimization is a topic of interest to both computer-aided design (CAD) specialists and computer science theoreticians. In the logic synthesis community, two-level logic minimization forms the foundation for more complex optimization procedures that have significant real-world impact. At the same time, the computational complexity of two-level logic minimization has posed challenges since the beginning of the field in the 1960s; indeed, some central questions have been resolved only within the last few years, and others remain open. This recent activity has classified some logic optimization problems of high practical relevance, such as finding the minimal sum-of-products (SOP) form and maximal term expansion and reduction. This paper surveys progress in the field with self-contained expositions of fundamental early results, an account of the recent advances, and some new classifications. It includes an introduction to the relevant concepts and terminology from computational complexity, as well a discussion of the major remaining open problems in the complexity of logic minimization.", "doi": "10.1109/TCAD.2005.855944", "issn": "0278-0070", "publisher": "IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems", "publication": "IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems", "publication_date": "2006-07-01", "series_number": "7", "volume": "25", "issue": "7", "pages": "1230-1246" }, { "id": "authors:9xmaq-tkq70", "collection": "authors", "collection_id": "9xmaq-tkq70", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161219-150257214", "type": "article", "title": "Simple extractors for all min-entropies and a new pseudorandom generator", "author": [ { "family_name": "Shaltiel", "given_name": "Ronen", "clpid": "Shaltiel-R" }, { "family_name": "Umans", "given_name": "Christopher", "clpid": "Umans-C" } ], "abstract": "A \"randomness extractor\" is an algorithm that given a sample from a distribution with sufficiently high min-entropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Min-entropy is a measure of the amount of randomness in a distribution.) We present a simple, self-contained extractor construction that produces good extractors for all min-entropies. Our construction is algebraic and builds on a new polynomial-based approach introduced by Ta-Shma et al. [2001b]. Using our improvements, we obtain, for example, an extractor with output length m = k/(log n)O(1/\u03b1) and seed length (1 + \u03b1)log n for an arbitrary 0 < \u03b1 \u2264 1, where n is the input length, and k is the min-entropy of the input distribution.\n\nA \"pseudorandom generator\" is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform. Our technique also gives a new way to construct pseudorandom generators from functions that require large circuits. Our pseudorandom generator construction is not based on the Nisan-Wigderson generator [Nisan and Wigderson 1994], and turns worst-case hardness directly into pseudorandomness. The parameters of our generator match those in Impagliazzo and Wigderson [1997] and Sudan et al. [2001] and in particular are strong enough to obtain a new proof that P = BPP if E requires exponential size circuits.Our construction also gives the following improvements over previous work:---We construct an optimal \"hitting set generator\" that stretches O(log n) random bits into s\u03a9(1) pseudorandom bits when given a function on log n bits that requires circuits of size s. This yields a quantitatively optimal hardness versus randomness tradeoff for both RP and BPP and solves an open problem raised in Impagliazzo et al. [1999].---We give the first construction of pseudorandom generators that fool nondeterministic circuits when given a function that requires large nondeterministic circuits. This technique also give a quantitatively optimal hardness versus randomness tradeoff for AM and the first hardness amplification result for nondeterministic circuits.", "doi": "10.1145/1059513.1059516", "issn": "0004-5411", "publisher": "Association for Computing Machinery", "publication": "Journal of the ACM", "publication_date": "2005-03", "series_number": "2", "volume": "52", "issue": "2", "pages": "172-216" } ]