Faster Combinatorial k-Clique Algorithms

TL;DR

This paper advances k-clique algorithms by developing a divide-and-conquer reduction from general k-cliques to triangle detection, leveraging Chan's and Yu's combinatorial strategies to shave log factors and beat the traditional $O(n^k)$ bound. It also extends combinatorial techniques beyond simple graphs to hypergraphs and triangle listing, achieving the first $o(n^k)$ algorithm for k-clique in hypergraphs and a near-subquadratic listing bound for triangles. By comparing combinatorial methods with matrix-multiplication-based approaches, the authors argue for broader generalizability and potential practical benefits. Overall, the work highlights how carefully designed combinatorial reductions and partition-based techniques can yield meaningful speedups in fundamental subgraph detection problems and their generalizations.

Abstract

Detecting if a graph contains a $k$-Clique is one of the most fundamental problems in computer science. The asymptotically fastest algorithm runs in time $O(n^{ωk/3})$, where $ω$ is the exponent of Boolean matrix multiplication. To date, this is the only technique capable of beating the trivial $O(n^k)$ bound by a polynomial factor. Due to this technique's various limitations, much effort has gone into designing "combinatorial" algorithms that improve over exhaustive search via other techniques. The first contribution of this work is a faster combinatorial algorithm for $k$-Clique, improving Vassilevska's bound of $O(n^{k}/\log^{k-1}{n})$ by two log factors. Technically, our main result is a new reduction from $k$-Clique to Triangle detection that exploits the same divide-and-conquer at the core of recent combinatorial algorithms by Chan (SODA'15) and Yu (ICALP'15). Our second contribution is exploiting combinatorial techniques to improve the state-of-the-art (even of non-combinatorial algorithms) for generalizations of the $k$-Clique problem. In particular, we give the first $o(n^k)$ algorithm for $k$-clique in hypergraphs and an $O(n^3/\log^{2.25}{n} + t)$ algorithm for listing $t$ triangles in a graph.