On Rigid Matrices and U-Polynomials

We introduce a class of polynomials, which we call U - polynomials , and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of U -pol...

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Vydané v:	Computational complexity Ročník 24; číslo 4; s. 851 - 879
Hlavní autori:	Alon, Noga, Cohen, Gil
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cham Springer International Publishing 01.12.2015
Predmet:	Algorithm Analysis and Problem Complexity Computational Mathematics and Numerical Analysis Computer Science small-bias sets Matrix rigidity unbalanced expanders 68Q17 68R05
ISSN:	1016-3328, 1420-8954
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Abstract	We introduce a class of polynomials, which we call U - polynomials , and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of U -polynomials, though their size is larger than desired. Furthermore, we give two alternative proofs for the fact that small-bias sets induce rigid matrices. Finally, we construct rigid matrices from unbalanced expanders, with essentially the same size as the construction via small-bias sets.
AbstractList	We introduce a class of polynomials, which we call U - polynomials , and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of U -polynomials, though their size is larger than desired. Furthermore, we give two alternative proofs for the fact that small-bias sets induce rigid matrices. Finally, we construct rigid matrices from unbalanced expanders, with essentially the same size as the construction via small-bias sets.
Author	Cohen, Gil Alon, Noga
Author_xml	– sequence: 1 givenname: Noga surname: Alon fullname: Alon, Noga organization: Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University – sequence: 2 givenname: Gil surname: Cohen fullname: Cohen, Gil email: gil.cohen@weizmann.ac.il organization: Department of Computer Science and Applied Mathematics, Weizmann Institute of Science
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Cites_doi	10.1145/1538902.1538904 10.1002/rsa.3240050203 10.1109/18.119713 10.1002/rsa.3240030308 10.1007/978-3-642-03685-9_26 10.1090/S0273-0979-06-01126-8 10.1109/CCC.2010.35 10.1109/SFCS.1995.492457 10.1016/0012-365X(88)90189-6 10.1016/S0020-0190(97)00190-7 10.1017/CBO9781139814782 10.1007/BF02311250 10.1137/0222053 10.1007/3-540-08353-7_135 10.1007/BF01303207 10.1109/FOCS.2009.44
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Constructing small-bias sets from algebraic-geometric codes. In Proceedings of the 50th Annual IEEE symposium on Foundations of Computer science (FOCS), 191–197. – reference: F. J. MacWilliams & N. J. A. Sloane (1977). The Theory of Error-Correcting Codes, Part II. North-Holland. – reference: ShokrollahiM.A.SpielmanD.StemannV.A remark on matrix rigidityInformation Processing Letters1997646283285160824010.1016/S0020-0190(97)00190-7 – reference: AlonN.GoldreichO.HåstadJ.PeraltaR.Simple Construction of Almost k-wise Independent Random VariablesRandom Structures and Algorithms1992332893040755.60002116484210.1002/rsa.3240030308 – reference: AlonN.RoichmanY.Random Cayley Graphs and ExpandersRandom Structures and Algorithms1994522712850798.05048126297910.1002/rsa.3240050203 – reference: Z. Dvir (2010). On Matrix Rigidity and Locally Self-Correctable Codes. In Proceedings of the 25th IEEE Conference on Computational Complexity (CCC), 291–298. – reference: V. Arvind & S. Srinivasan (2010). 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