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We present a new combinatorial algorithm for triangle finding and Booleanmatrix multiplication that runs in $\hat{O}(n^3/\log^4 n)$ time, where the$\hat{O}$ notation suppresses poly(loglog) factors. This improves the previousbest combinatorial algorithm by Chan that runs in $\hat{O}(n^3/\log^3 n)$ time.Our algorithm generalizes the divide-and-conquer strategy of Chan's algorithm.Moreover, we propose a general framework for detecting triangles in graphs andcomputing Boolean matrix multiplication. Roughly speaking, if we can find the"easy parts" of a given instance efficiently, we can solve the whole problemfaster than $n^3$.

An Improved Combinatorial Algorithm for Boolean Matrix Multiplication