# Faster Combinatorial k-Clique Algorithms

CoRR（2024）

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^ω k/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-1n) 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^2.25n + t) algorithm for
listing t triangles in a graph.

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