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 nnode 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.