Some Lower Bound Results for Set-Multilinear Arithmetic Computations

TL;DR

This paper investigates the structure of set-multilinear arithmetic models, establishing lower bounds for restricted models and demonstrating their computational limitations, especially for the Permanent polynomial.

Contribution

It introduces new lower bounds for restricted set-multilinear models and explores their computational power relative to other models, extending and raising open questions in the field.

Findings

Set-multilinear circuits can be efficiently depth-reduced.

Narrow set-multilinear ABPs computing PER_n require exponential size.

Set-multilinear branching programs are exponentially more powerful than interval multilinear circuits.

Abstract

In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of showing lower bound results. We define some natural restrictions of these models for which we are able to show lower bound results. Some of our results extend existing lower bounds, while others are new and raise open questions. More specifically, our main results are the following: (1) We observe that set-multilinear arithmetic circuits can be transformed into shallow set-multilinear circuits efficiently, similar to depth reduction results of [VSBR83,RY08] for more general commutative circuits. As a consequence, we note that polynomial size set-multilinear circuits have quasi-polynomial size set-multilinear branching programs. We show that \emph{narrow} set-multilinear ABPs (with a restricted number of set types) computing the Permanent polynomial $\mathrm{PER}_n$ require $2^{n^{\Omega(1)}}$ size. A similar result for general set-multilinear ABPs appears difficult as it would imply that the Permanent requires superpolynomial size set-multilinear circuits. It would also imply that the noncommutative Permanent requires superpolynomial size noncommutative arithmetic circuits. (2) Indeed, we also show that set-multilinear branching programs are exponentially more powerful than \emph{interval} multilinear circuits (where the index sets for each gate is restricted to be an interval w.r.t.\ some ordering), assuming the sum-of-squares conjecture. This further underlines the power of set-multilinear branching programs. (3) Finally, we consider set-multilinear circuits with restrictions on the number of proof trees of monomials computed by it, and prove exponential lower bounds results. This raises some new lower bound questions.