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Efficiently-Verifiable Strong Uniquely Solvable Puzzles and Matrix Multiplication

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Computing and Combinatorics (COCOON 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14423))

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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 bounds on the matrix multiplication exponent \(\omega \) that infinite families of SUSPs can. We construct, by computer search, larger SUSPs than known for small width. This, combined with our tighter analysis, strengthens the upper bound on \(\omega \) from 2.66 to 2.505 obtainable via this computational approach, nearing the handcrafted constructions of Cohn-Umans.

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Notes

  1. 1.

    Implementations of our algorithms, along with a tool for verifying simplifiable SUSPs, are publicly available at https://bitbucket.org/paraphase/matmult-v2.

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Acknowledgments

The second author’s work was funded in part by the Union College Summer Research Fellows Program. Both authors acknowledge contributions from other student researchers to various aspects of this research program. We thank our anonymous reviewers for their helpful suggestions.

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Authors and Affiliations

  1. Department of Computer Science, Union College, Schenectady, NY, USA

    Matthew Anderson & Vu Le

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  1. Matthew Anderson
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  2. Vu Le
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Correspondence to Matthew Anderson .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. University of Delaware, Newark, DE, USA

    Guangmo Tong

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Anderson, M., Le, V. (2024). Efficiently-Verifiable Strong Uniquely Solvable Puzzles and Matrix Multiplication. In: Wu, W., Tong, G. (eds) Computing and Combinatorics. COCOON 2023. Lecture Notes in Computer Science, vol 14423. Springer, Cham. https://doi.org/10.1007/978-3-031-49193-1_4

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  • DOI: https://doi.org/10.1007/978-3-031-49193-1_4

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  • Online ISBN: 978-3-031-49193-1

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