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A local search based restart evolutionary algorithm for finding triple product property triples

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Abstract

As a mean to bound the exponent ω of the matrix multiplication, the group-theoretic approach to fast matrix multiplication was first introduced by Cohn and Umans in 2003. This involves a knotty problem, i.e., finding three subsets of a given group satisfying the so-called triple product property such that the product of their sizes is as large as possible. For this challenge problem, exact or randomized heuristic algorithms have been proposed, which are either time-consuming or ineffective on groups with large order. This paper proposes to use an evolutionary algorithm to solve the above problem. In the proposed algorithm, a local search and a restart strategy are employed to enhance the exploitation and exploration ability of the algorithm, respectively. The proposed approach is tested on a large number of nonabelian groups with order from 6 to 100. Experimental results show that the new algorithm can obtain a good tradeoff among effectiveness, robustness and efficiency. Especially, this approach is effective for nonabelian groups with order larger than 36, and obtains three subsets for each of these groups, which satisfy the triple product property and the product of whose sizes reaches the best found so for. Most importantly, we find by using the proposed algorithm 12 groups which would be promising to prove a nontrivial upper bound on the exponent ω of the matrix multiplication.

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Notes

  1. GAP - Groups, Algorithms, Programming - a system for computational discrete algebra, is available at http://www.gap-system.org/

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Acknowledgments

The authors are grateful to the anonymous reviewers for their insightful comments. We thank Haiping Hu and Xinsheng Lai for their helpful discussions.

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Authors and Affiliations

  1. School of Data and Computer Science & Collaborative Innovation Center of High Performance Computing, Sun Yat-sen University, Guangzhou, 510006, People’s Republic of China

    Yi Xiang, Yuren Zhou & Zefeng Chen

Authors
  1. Yi Xiang
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  2. Yuren Zhou
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  3. Zefeng Chen
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Corresponding author

Correspondence to Yuren Zhou.

Additional information

This paper is supported by the National Natural Science Foundation of China (Grant nos. 61472143, 61773410 and 61673403), the Scientific Research Special Plan of Guangzhou Science and Technology Program (Grant no. 201607010045) and the Excellent Graduate Student Innovation Program at Collaborative Innovation Center of High Performance Computing.

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Xiang, Y., Zhou, Y. & Chen, Z. A local search based restart evolutionary algorithm for finding triple product property triples. Appl Intell 48, 2894–2911 (2018). https://doi.org/10.1007/s10489-017-1118-6

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