Abstract

A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial-time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial-time complexity $${n^{O({\rm log}\,n)}}$$ . In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of set-multilinear depth-3 circuits. The prior results for the sum of constantly many set-multilinear depth-3 circuits were only slightly better than brute force, i.e., exponential time. Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension.

Highlights

  • Polynomial Identity Testing (PIT) is the problem of testing whether a given n-variate polynomial is identically zero or not

  • D, w, in timeO(log ndw) one can construct a hitting-set for all n-variate polynomials of individual degree d, that can be computed by a sum of two read-once oblivious arithmetic branching program (ABP) (ROABP) of width w

  • As a by-product, we show that low support concentration can be achieved even when we have a sum of matrix polynomials, each computed by an ROABP

Read more

Summary

Introduction

Polynomial Identity Testing (PIT) is the problem of testing whether a given n-variate polynomial is identically zero or not. An efficient deterministic solution for PIT is known only for very restricted input models, for example, sparse polynomials [5, 19], constant fan-in depth-3 (ΣΠΣ) circuits [7, 18, 17, 16, 27, 28], set-multilinear circuits [22, 10, 4], read-once oblivious ABP (ROABP) [22, 12, 9, 3] This lack of progress is not surprising: Gupta et al [13] showed that a polynomial time test for depth-3 circuits would imply a sub-exponential time test for general circuits. We only require that the weight of a coefficient is greater than the weight of the basis coefficients that it depends on

Notation
Arithmetic branching programs
Read-once oblivious arithmetic branching programs
Whitebox Identity Testing
Equivalence of two ROABPs
Sum of constantly many ROABPs
Blackbox Identity Testing
Sum of ROABPs
Concentration in matrix polynomials
Low Support Concentration in ROABPs
Discussion
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call