An Improved Combinatorial Algorithm for Boolean Matrix Multiplication

TL;DR

This paper introduces a faster combinatorial algorithm for Boolean matrix multiplication and triangle detection, improving previous methods by leveraging a generalized divide-and-conquer framework that efficiently identifies easier subproblems.

Contribution

It presents a new algorithm that improves the time complexity of combinatorial Boolean matrix multiplication and triangle finding, extending Chan's approach with a general framework.

Findings

Achieves $ ilde{O}(n^3/ ext{log}^4 n)$ runtime for Boolean matrix multiplication.

Generalizes divide-and-conquer strategy for triangle detection.

Provides a framework for identifying and solving easier subproblems efficiently.

Abstract

We present a new combinatorial algorithm for triangle finding and Boolean matrix multiplication that runs in $\hat{O}(n^3/\log^4 n)$ time, where the $\hat{O}$ notation suppresses poly(loglog) factors. This improves the previous best combinatorial algorithm by Chan that runs in $\hat{O}(n^3/\log^3 n)$ time. Our algorithm generalizes the divide-and-conquer strategy of Chan's algorithm. Moreover, we propose a general framework for detecting triangles in graphs and computing Boolean matrix multiplication. Roughly speaking, if we can find the "easy parts" of a given instance efficiently, we can solve the whole problem faster than $n^3$.