Upgrading Subgroup Triple-Product-Property Triples

Abstract

In 2003, Cohn and Umans introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group satisfying the Triple Product Property (TPP) as a means to bound the exponent of matrix multiplication. Recently, Hedtke and Murthy discussed several methods to find TPP triples. Because the search space for subset triples is too large, it is only possible to focus on subgroup triples. We present methods to upgrade a given TPP triple to a bigger TPP triple. If no upgrade is possible, we use reduction methods (based on random experiments and heuristics) to create a smaller TPP triple that can be used as input for the upgrade methods. If we apply the upgrade process for subset triples after one step with the upgrade method for subgroup triples for the known maximal subgroup TPP triples in groups of order up to 1,000, we achieve an enlargement of the triple size of 100% in the best case. Further, we test the upgrade process with all examples from the 2003 and 2005 papers from Cohn et al. and are able to increase the triple size by 595% in the best case (in the group D 5 6 ).