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) · n k O(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) in general, and time n k O(k 2 ) +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.

Hitting Sets and Reconstruction for Dense Orbits in $\text{VP}_e$ and $ΣΠΣ$ Circuits

“…Minahan and Volkovich obtained a polynomial-sized hitting set for the class, which led to a similar improvement in the running time of the reconstruction algorithm [MV18]. Anderson, van Melkebeek and Volkovich constructed a hitting set of size n k O(k) +O(k log n) for read-k formulas [AvMV15]. All these results work in a slightly stronger model in which we allow to label leaves with univariate polynomials, of polynomial degree, such that every variable appears in at most one polynomial, or with sparse polynomials on disjoint sets of variables.…”

Characterizing Arithmetic Read-Once Formulae

“…+ P k is uniquely defined by its O(k)-variate restrictions to a typical assignment. The result was recently generalized in [AvMV14] showing that a polynomial computed by multilinear read-k is uniquely defined by its k O(k) -variate restrictions to a typical assignment. In [SV14], it was shown how to efficiently reconstruct a (single) read-once formula given the set of its three-variate restrictions to a typical assignment.…”