A generalization of recent group-theoretic matrix multiplication algorithms to an analogue of the theory of partial matrix multiplication is presented. We demonstrate that the added flexibility of this approach can in some cases improve upper bounds on the exponent of matrix multiplication yielded by group-theoretic full matrix multiplication. The group theory behind our partial matrix multiplication algorithms leads to the problem of maximizing a quantity representing the "fullness" of a given partial matrix pattern. This problem is shown to be NP-hard, and two algorithms, one optimal and another non-optimal but polynomial-time, are given for solving it.

An improvement upon the naive O(n3) algorithm for matrix multiplication was first presented by Strassen, obtaining the result in only O(n2.81) field operations [5]. This raises the question of what the best possible exponent k such that matrix multiplication can be carried out in at most O(nk) time is. Clearly k ≥ 2, since n2 is the size of the output. It is believed that the optimal k is exactly equal to 2, but this has yet to be proven [3]. Presently, the fastest known algorithm computes the product of two matrices in at most O(n2.38) operations. Recent work by Cohn and Umans indicates a possible path to proving that the obvious lower bound of 2 is tight, namely an approach using techniques from group theory and representation theory [3]. Their proposed algorithm is analogous to the way that the Discrete Fourier Transform (DFT) computes the product of two polynomials by embedding them in a cyclic group algebra over the complex numbers, then computing the pointwise product of vectors in the appropriate complex vector space. Since abelian groups cannot yield the properties necessary to achieve k = 2, it is instead necessary to embed the matrices in a nonabelian group algebra. This complicates multiplication in the Fourier domain; rather than being simply pointwise vector multiplication, it becomes multiplication of block-diagonal matrices with block sizes determined by the irreducible representations of the group.

Abstract Recent work,has shown,that fast matrix multiplication,algorithms,can be constructed by embedding the two input matrices into a group algebra, ap- plying a generalized discrete Fourier transform, and performing the multi- plication in the Fourier basis. Developing,an embedding,that yields a ma- trix multiplication,algorithm,with running,time faster than naive matrix multiplication,leads to interesting combinatorial,problems,in group,theory. The crux of such an embedding, after a group G has been chosen, lies in finding a triple of subsets of G that satisfy a certain algebraic relation. I show,how,the process of finding such subsets can in some,cases be greatly simplified by considering,the action of the group,G on an appropriate,set X. In particular, I focus on groups acting on regularly branching trees. Contents