Monochromatic Triangles, Intermediate Matrix Products, and Convolutions

The most studied linear algebraic operation, matrix multiplication, has surprisingly fast $O(n^\omega)$ time algorithms for $\omega<2.373$. On the other hand, the $(\min,+)$ matrix product which is at the heart of many fundamental graph problems such as APSP, has received only minor improvements over its brute-force cubic running time and is widely conjectured to require $n^{3-o(1)}$ time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems. For instance, the Min-Max matrix product, the Minimum Witness matrix product, APSP in directed unweighted graphs and determining whether an edge-colored graph contains a monochromatic triangle, can all be solved in $\tilde O(n^{(3+\omega)/2})$ time. A similar phenomenon occurs for convolution problems, where analogous intermediate problems can be solved in $\tilde O(n^{1.5})$ time. Can one improve upon the running times for these intermediate problems, in either the matrix product or the convolution world? Or, alternatively, can one relate these problems to each other and to other key problems in a meaningful way? This paper makes progress on these questions by providing a network of fine-grained reductions. We show for instance that APSP in directed unweighted graphs and Minimum Witness product can be reduced to both the Min-Max product and a variant of the monochromatic triangle problem. We also show that a natural convolution variant of monochromatic triangle is fine-grained equivalent to the famous 3SUM problem. As this variant is solvable in $O(n^{1.5})$ time and 3SUM is in $O(n^2)$ time (and is conjectured to require $n^{2-o(1)}$ time), our result gives the first fine-grained equivalence between natural problems of different running times.