Towards Optimal Output-Sensitive Clique Listing or: Listing Cliques from Smaller Cliques

Abstract

We study the problem of nding and listing -cliques in an -edge, -vertex graph, for constant ≥ 3. This is a fundamental problem of both theoretical and practical importance. Our rst contribution is an algorithmic framework for nding -cliques that gives the rst improvement in 19 years over the old runtimes for 4 and 5-clique nding, as a function of [Eisenbrand and Grandoni, TCS’04]. With the current bounds on matrix multiplication, our algorithms run in (1.66) and (2.06) time, respectively, for 4-clique and 5-clique nding. Our main contribution is an output-sensitive algorithm for listing -cliques, for any constant ≥ 3. We complement the algorithm with tight lower bounds based on standard ne-grained assumptions. Previously, the only known conditionally optimal outputsensitive algorithms were for the case of 3-cliques given by Björklund, Pagh, Vassilevska W. and Zwick [ICALP’14]. If the matrix multiplication exponent is 2, and if the number of -cliques is large enough, the running time of our algorithms is ˜

min{ 1 −2 1− 2 (−2) , 2 −1 1− 2 (−1) }

, and this is tight under the Exact--Clique Hypothesis. This running time naturally extends the running time obtained by Björklund, Pagh, Vassilevska W. and Zwick for = 3. Our framework is very general in that it gives -clique listing algorithms whose running times can be measured in terms of the number of ℓ-cliques Δℓ in the graph for any 1 ≤ ℓ < . This generalizes the typical parameterization in terms of (the number of 1-cliques) and (the number of 2-cliques). If is 2, and if the size of the output, Δ , is su ciently large, then for every ℓ < , the running time of our algorithm for listing -cliques is ˜

Δ 2 ℓ(−ℓ) ℓ Δ 1− 2 (−ℓ)