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(n ω ) 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, iñ O(n ω ) time with high probability. The algorithm can be derandomized in some cases: we show how to detect a diamond (or its complement) in deterministicÕ(n ω ) time. Our approach substantially improves on prior work. For instance, the previous best algorithm for C 4 detection ran in O(n 3.3 ) time, and for diamond detection in O(n 3 ) time. • For sparse graphs with m edges, the best known triangle finding algorithm runs in O(m 2ω/(ω+1) ) ≤ O(m 1.41 ) time. We give a randomizedÕ(m 2ω/(ω+1) ) time algorithm (analogous to the best known for triangle finding) for finding any induced four-node subgraph other than C 4 , K 4 and their complements. In the case of diamond detection, we also design a deterministicÕ(m 2ω/(ω+1) ) time algorithm. For C 4 or its complement, we give randomized O(m (4ω−1)/(2ω+1) ) ≤ O(m 1.48 ) time finding algorithms. These algorithms substantially improve on prior work. For instance, the best algorithm for diamond detection ran in O(m 1.5 ) time.
Williams, Wang, Williams, and Yu REF gave O(n ω ) time algorithms for the induced subgraph isomorphism problems for four vertex pattern graphs, except for I 4 and K 4 .
Finding Four-Node Subgraphs in Triangle Time
SODA
Mathematics Computer Science