Improved polynomial identity testing for read-once formulas

Amir Shpilka*, Ilya Volkovich

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

An arithmetic read-once formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are {+,×} and such that every input variable labels at most one leaf. In this paper we study the problems of giving deterministic identity testing and reconstruction algorithms for ROFs. Our main result is an nscript O sign(k+log n) time deterministic algorithm for checking whether a black box holding the sum of k n-variate ROFs computes the zero polynomial. In other words, we provide a hitting set of size nscript O sign(k+log n) for the sum of k ROFs. This result greatly improves [27] where an n script O sign(k2+√n) algorithm was given for the problem. Using our new results we obtain a deterministic reconstruction algorithms for read-once formulas that runs in time script O sign(log n). In fact, our results also hold for the more general model of preprocessed read-once formulas that we define in this paper. In this model we are allowed to replace each variable xi with a polynomial Ti (xi ). Our techniques are very close to the techniques in [27]. The main difference is that we obtain several tighter versions of the tools first used there. In particular we obtain a better version of the hardness of representation approach which was first used in [27]. This technique can be thought of as a very explicit way of transforming (mild) hardness of a very structured polynomial to an identity testing algorithm.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings
Pages700-713
Number of pages14
DOIs
StatePublished - 2009
Externally publishedYes
Event12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009 - Berkeley, CA, United States
Duration: 21 Aug 200923 Aug 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5687 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009
Country/TerritoryUnited States
CityBerkeley, CA
Period21/08/0923/08/09

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