Ant colony optimization for triple product property triples to fast matrix multiplication

Abstract

The group-theoretic approach to fast matrix multiplication has been introduced by Cohn and Umans (proceedings of 44th annual IEEE symposium on foundations of computer science, IEEE Computer Society, 2003). In this approach, there is a challenging problem. It is to find 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 challenging problem, some brute-force algorithm has been proposed, which is exact but time-consuming. The ant colony optimization is a randomized heuristic algorithm and finds extensive applications in many fields. We use it to solve the problem of searching for three subsets of a given group such that they satisfy the triple product property and the product of their sizes reaches the maximum. Experimental results show that the ant colony optimization is efficient for this problem. Using this approach, we obtain three subsets for each nonabelian group of order not larger than 44 which satisfy the triple product property and the product of whose sizes reaches the best value found so far.