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
Summary
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
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