Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles

TL;DR

This paper develops computational methods to identify strong uniquely solvable puzzles, which are linked to faster matrix multiplication algorithms, and provides bounds and constructions for small widths.

Contribution

It introduces constraint-based algorithms for verifying and searching for strong USPs, producing new bounds and constructions for small widths, and explores their implications for matrix multiplication complexity.

Findings

Bounds on maximum size of strong USPs for width ≤ 5

Construction of larger puzzles than previous work for small widths

Improved upper bounds on strong USP size for width ≤ 12

Abstract

Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras. In subsequent work with Kleinberg and Szegedy, they connected this to the search for combinatorial objects called strong uniquely solvable puzzles (strong USPs). We begin a systematic computer-aided search for these objects. We develop and implement constraint-based algorithms build on reductions to $\mathrm{SAT}$ and $\mathrm{IP}$ to verify that puzzles are strong USPs, and to search for large strong USPs. We produce tight bounds on the maximum size of a strong USP for width $k \le 5$, construct puzzles of small width that are larger than previous work, and improve the upper bounds on strong USP size for $k \le 12$. Although our work only deals with puzzles of small-constant width, the strong USPs we find imply matrix multiplication algorithms that run in $O(n^\omega)$ time with exponent $\omega \le 2.66$. While our algorithms do not beat the fastest algorithms, our work provides evidence and, perhaps, a path to finding families of strong USPs that imply matrix multiplication algorithms that are more efficient than those currently known.