Counting and detecting small subgraphs via equations and matrix multiplication

Abstract

We present a general technique for detecting and counting small subgraphs. It consists in forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed size in a graph. The combinations can be efficiently computed by rectangular matrix multiplication. Our two main results utilizing the technique are as follows. Let H be a fixed graph with k vertices and an independent set of size s. Detecting if an u-vertex graph contains a (non- necessarily induced) subgraph isomorphic to H can be done in time O(nk-s + nω (⌈k-s)/2⌉,1, ⌊ (k-s)/2⌋)), where u ω (p,q,r) is the exponent of fast arithmetic matrix multiplication of np × nq matrix by an nq × nr matrix.2. When s = 2, counting the number of (non-necessarily induced) subgraphs isomorphic to H can be done in the same time, i.e., in time O(nk-2 + nω (⌈(k-2)/2⌉,1, ⌊ (k-2)/2⌋)) (This improves for s = 2 on a counting algorithm of Vassilevska and Williams, running in time O(nk-s+3).) It follows in particular that we can count the number of subgraphs isomorphic to any H on four vertices that is not K4 in time O(nω), where ω = ω ( 1,1,1) is known to be smaller than 2.376. Similarly, we can count the number of subgraphs isomorphic to any H on five vertices that is not K5 in time O(nω (2,1,1))where ω (2,1,1) is known to be smaller than 3.334.