Finding Four-Node Subgraphs in Triangle Time

Abstract

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.

Revisiting Decomposition by Clique Separators

“…It is unlikely that the problem can be solved in o(n α )-time by Theorem 6 (recall that the two problems of triangle detection and matrix multiplication are assumed to be equivalent [43,42]). Finally, we proved in Theorem 9 that for every graph G with bounded clique-number ω, the clique minimal separator decomposition of G can be computed in O(T (G) + ω 2 n)-time where T (G) here denotes the time needed to compute a minimal triangulation of G. We conjecture that in fact, it can be computed in O(ω O(1) (n + m))-time.…”

“…Altogether, this is hint that ourÕ(n α )-time clique minimal separator decomposition algorithm is optimal up to polylogarithmic factors -assuming the computational equivalence between triangle detection and matrix multiplication [43,42].…”

“…This combined with standard complexity assumptions -such as the Strong Exponential-Time Hypothesis [31] -allows one to prove conditional time complexity lower bounds for many problems in P, and unsuspected relationships between these problems. In our case we will assume, like in [42], the computational equivalence between Triangle Detection and Matrix Multiplication. This equivalence has been proved in [43] for combinatorial algorithms (i.e., not using algebraic methods).…”

Detecting and Counting Small Pattern Graphs

“…In a subsequent paper, after the submission of this work, Vassilevska Williams et al [21] presented several improved upper bounds on the detection of induced subgraphs on four vertices and even some induced subgraphs on five vertices (e.g., the chair and 4-pan). By using a polynomial testing framework similar to that in this paper, they have shown in particular that any induced subgraph on four vertices, with the exception of K 4 and the independent set on four vertices, can be detected by a randomized algorithm in O(n ω ) time.…”

“…By using a polynomial testing framework similar to that in this paper, they have shown in particular that any induced subgraph on four vertices, with the exception of K 4 and the independent set on four vertices, can be detected by a randomized algorithm in O(n ω ) time. The improved upper time bounds presented in [21] are noncombinatorial as they rely on fast matrix multiplication.…”