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 \(O(n^{\omega k/3})\) , where \(\omega \) 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. Specifically we introduce the first non-trivial clique listing algorithms: A general algorithm for detecting cliques in hypergraphs that also improves the state of the art for detection, and an additional algorithm for the specific case of triangle listing with improved bounds.
