Efficiently-Verifiable Strong Uniquely Solvable Puzzles and Matrix Multiplication

TL;DR

This paper introduces a new subclass of strong uniquely solvable puzzles called simplifiable SUSPs, which are efficiently verifiable and help improve the upper bound on the matrix multiplication exponent to 2.505 through computational methods.

Contribution

It defines and analyzes simplifiable SUSPs, demonstrating their efficiency and potential to match bounds of infinite SUSPs, and improves the upper bound on matrix multiplication exponent via computer search.

Findings

Achieved a tighter upper bound on matrix multiplication exponent (ω) at 2.505.

Constructed larger SUSPs for small width through computational search.

Showed simplifiable SUSPs are efficiently verifiable, unlike general SUSPs.

Abstract

We advance the Cohn-Umans framework for developing fast matrix multiplication algorithms. We introduce, analyze, and search for a new subclass of strong uniquely solvable puzzles (SUSP), which we call simplifiable SUSPs. We show that these puzzles are efficiently verifiable, which remains an open question for general SUSPs. We also show that individual simplifiable SUSPs can achieve the same strength of bounds on the matrix multiplication exponent $\omega$ that infinite families of SUSPs can. We report on the construction, by computer search, of larger SUSPs than previously known for small width. This, combined with our tighter analysis, strengthens the upper bound on the matrix multiplication exponent from $2.66$ to $2.505$ obtainable via this computational approach, and nears the results of the handcrafted constructions of Cohn et al.