Verifying the Product of Generalized Boolean Matrix Multiplication and Its Applications to Detect Small Subgraphs

Wing-Kai Hon, Meng-Tsung Tsai, Hung-Lung Wang

Abstract: Given three n by n integer matrices A, B, and P, determining whether the product AB equals P can be done in randomized \(O(n^2)\) time by Freivalds’ algorithm. In this paper, we consider some generalized Boolean matrix multiplication \(A B = P_f\), which is defined to be setting the entry \(p_{ij}\) of \(P_f\) for \(i, j \in [n]\) as the value of a given function f of the entries on the ith row of A and jth column of B. We show that, for a family of functions f, it takes deterministic \(O(n^2)\) time to verify whether the generalized product \(P_f\) contains only False entries. Then, we present how to apply such a result to detect small subgraphs efficiently, including: