Lower bounds for Combinatorial Algorithms for Boolean Matrix Multiplication

TL;DR

This paper establishes new lower bounds for combinatorial algorithms solving Boolean Matrix Multiplication by introducing models extending previous work and leveraging special graph structures and randomization.

Contribution

It proposes two combinatorial models for BMM and proves significant lower bounds, advancing understanding of algorithmic limitations in this area.

Findings

Lower bound of Ω(n^3 / 2^{O(√log n)}) for a relaxed model

Lower bound of Ω(n^{7/3} / 2^{O(√log n)}) for a more general model

Use of (r,t)-graphs and randomization to construct hard instances

Abstract

In this paper we propose models of combinatorial algorithms for the Boolean Matrix Multiplication (BMM), and prove lower bounds on computing BMM in these models. First, we give a relatively relaxed combinatorial model which is an extension of the model by Angluin (1976), and we prove that the time required by any algorithm for the BMM is at least $\Omega(n^3 / 2^{O( \sqrt{ \log n })})$. Subsequently, we propose a more general model capable of simulating the "Four Russians Algorithm". We prove a lower bound of $\Omega(n^{7/3} / 2^{O(\sqrt{ \log n })})$ for the BMM under this model. We use a special class of graphs, called $(r,t)$-graphs, originally discovered by Rusza and Szemeredi (1978), along with randomization, to construct matrices that are hard instances for our combinatorial models.