Paper Info
Title
Deterministic polynomial identity tests for multilinear bounded-read formulae
Abstract
We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula, each variable occurs only a constant number of times, and each subformula computes a multilinear polynomial. Our algorithm runs in time $${s^{O(1)}\\cdot n^{k^{O(k)}}}$$sO(1)·nkO(k) , where s denotes the size of the formula, n denotes the number of variables, and k bounds the number of occurrences of each variable. Before our work, no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time $${n^{k^{O(k)} + O(k \\log n)}}$$nkO(k)+O(klogn) in general, and time $${n^{k^{O(k^2)} + O(kD)}}$$nkO(k2)+O(kD) for depth D. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae and for multilinear depth-four formulae.
Year
DOI
Venue
2015
10.1007/s00037-015-0097-4
Computational Complexity
Keywords
Field
DocType
Derandomization, identity testing, arithmetic circuits, bounded-depth circuits, 12Y05, 68Q25
Discrete mathematics,Binary logarithm,Arithmetic circuits,Combinatorics,Polynomial,Identity testing,Multilinear polynomial,Deterministic algorithm,Multilinear map,Mathematics,Bounded function
Journal
Volume
Issue
ISSN
24
4
1420-8954
Citations 
PageRank 
References 
5
0.44
33
Authors
3
Name
Order
Citations
PageRank
Matthew Anderson1332.67
Dieter van Melkebeek248732.80
Ilya Volkovich31458.64