Improved Polynomial Identity Testing for Read-Once Formulas

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. A preprocessed ROF (PROF for short) is a ROF in which we are allowed to replace each variable x i with a univariate polynomial T i (x i ). In this paper we study the problems of giving deterministic identity testing and reconstruction algorithms for preprocessed ROFs. In particular we obtain the following results. We give an (nd)O(log n) black-box polynomial identity-testing algorithm for PROFs in n variables of individual degrees at most d (i.e. each T i (x i ) is of degree at most d). This improves and generalizes the previous n2. Given k PROFs in n variables of individual degrees at most d we give a deterministic (non black-box) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k) . This result improves and extends the previous results of [SV08].3. Combining the two results above we obtain an (nd) O(k+log n) time deterministic algorithm for checking whether a black box holding the sum of k n-variate PROFs computes the zero polynomial. In other words, we provide a hitting set of size (nd) O(k+log n) for the sum of k PROFs. This result greatly improves and extends [SV08] where an n O(k 2 + √ n) algorithm was given for identity testing the sum of k ROFs. We give an (nd)O(D+k) time deterministic black-box identity testing algorithm for the sum of k PROFs of depth D and individual degrees at most d, which improves the n O(D+k 2 ) algorithm of [SV08] for depth D ROFs.As a corollary of the results above we obtain an n O(k) time deterministic identity testing algorithm for multilinear depth-3 ΣΠΣ(k) circuits. This matches the non black-box algorithm of [KS07] for the multilinear case. In fact the same result also holds for preprocessed-multilinear-ΣΠΣ(k) circuits which are depth-4 circuits of a restricted form.In addition to the above results we obtain a deterministic reconstruction algorithms for preprocessed read-once formulas. The running time of the algorithm is (nd) O(log n) for general preprocessed read-once formulas, of individual degrees at most d, and (nd) O(D) for depth D preprocessed read-once formulas of individual degrees at most d.Most of our results are obtained using hardness of representation approach which was first used in [SV08]. 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.

Arithmetic Circuits: A survey of recent results and open questions

“…The first is by [Sax08] that gave a polynomial time PIT for the so called diagonal circuits, based on the ideas of [RS05]. The second result is by [KMSV10] that gave a PIT algorithm for depth-4 multilinear circuits with bounded top fan-in, based on ideas from [KS08] and [SV09]. Finally, we present the algorithm of [SV08,SV09] for identity testing of sums of read-once formulas that strengthen some of the results for depth-3 circuits and that influenced [KMSV10].…”

On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

“…This can be done by replacing, for each i and j, the j-th occurrence of x i with a new variable x i,j . Now, using PIT algorithm for read-once formulas [SV08,SV09], check whether this formula is zero or not. If it is zero then the original formulas was also zero and we are done.…”

“…Thus, we somehow have to find a way of verifying whether a linear function is a factor of a multilinear formula. Notice that as we start with a read-once formula for which PIT is known [SV08,SV09], we can assume that we know many inputs on which the formula does not vanish. One may hope that before replacing x i,j with x i we somehow managed to obtain inputs that will enable us to verify whether x i − x i,j is a factor of the formula or not.…”

“…In [SV08,SV09] justifying assignments were used in conjunction with new PIT algorithms in order to obtain deterministic quasi-polynomial time interpolation algorithms for read-once formulas. We rely on the ideas from [SV09] for obtaining justifying assignments from PIT algorithms.…”

“…If f is computed by sum of k Preprocessed Read-Once arithmetic formulas of individual degrees at most d (see [SV09]), then there is an (nd) O(k 2 ) non black-box algorithm for computing the indecomposable factors of f and an (nd) O(k 2 +log n) black-box algorithm for the problem.…”