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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Vydáno v:Computational complexity Ročník 24; číslo 4; s. 851 - 879
Hlavní autoři: Alon, Noga, Cohen, Gil
Médium: Journal Article
Jazyk:angličtina
Vydáno: Cham Springer International Publishing 01.12.2015
Témata:
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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Popis
Shrnutí: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.
ISSN:1016-3328
1420-8954
DOI:10.1007/s00037-015-0112-9