Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

TL;DR

This paper presents the first polynomial-time whitebox and a quasi-polynomial blackbox identity testing algorithms for sums of a constant number of read-once oblivious arithmetic branching programs, advancing algebraic circuit identity testing.

Contribution

It introduces the first efficient algorithms for identity testing of sums of ROABPs, extending techniques to handle sums of two ROABPs using novel concepts.

Findings

Whitebox algorithm runs in polynomial time.

Blackbox algorithm runs in quasi-polynomial time.

Techniques involve low evaluation dimension and basis isolating weights.

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 $n^{O(\log n)}$. 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 (or partial derivatives).