  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.
