Finding four-node subgraphs in triangle time

We present new algorithms for finding induced four-node subgraphs in a given graph, which run in time roughly that of detecting a clique on three nodes (i.e., a triangle). • The best known algorithms for triangle finding in an n-node graph take O(nω) time, where ω < 2.373 is the matrix multiplication exponent. We give a general randomized technique for finding any induced four-node subgraph, except for the clique or independent set on 4 nodes, in Õ(nω) time with high probability. The algorithm can be derandomized in some cases: we show how to detect a diamond (or its complement) in deterministic Õ(nω) time. Our approach substantially improves on prior work. For instance, the previous best algorithm for C4 detection ran in O(n3.3) time, and for diamond detection in O(n3) time. • For sparse graphs with m edges, the best known triangle finding algorithm runs in O(m2ω/(ω+1)) ≤ O(m1.41) time. We give a randomized Õ(m2ω/(ω+1)) time algorithm (analogous to the best known for triangle finding) for finding any induced four-node subgraph other than C4, K4 and their complements. In the case of diamond detection, we also design a deterministic Õ(m2ω/(ω+1)) time algorithm. For C4 or its complement, we give randomized Õ(m(4ω−1)/(2ω+1)) ≤ O(m1.48) time finding algorithms. These algorithms substantially improve on prior work. For instance, the best algorithm for diamond detection ran in O(m1.5) time.