An Improved Combinatorial Algorithm for Boolean Matrix Multiplication

We present a new combinatorial algorithm for triangle finding and Boolean matrix multiplication that runs in hatO(n3/log4 n) time, where the hatO notation suppresses poly(loglog) factors. This improves the previous best combinatorial algorithm by Chan that runs in hatO(n3/log3 n) time. Our algorithm generalizes the divide-and-conquer strategy of Chan's algorithm. Moreover, we propose a general framework for detecting triangles in graphs and computing Boolean matrix multiplication. Roughly speaking, if we can find the "easy parts" of a given instance efficiently, we can solve the whole problem faster than n3.