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

Mina Dalirrooyfard, Surya Mathialagan, Virginia Vassilevska Williams, Yinzhan Xu

Abstract: We study the problem of finding and listing k-cliques in an m-edge, n-vertex graph, for constant k≥ 3. This is a fundamental problem of both theoretical and practical importance.Our first contribution is an algorithmic framework for finding k-cliques that gives the first improvement in 19 years over the old runtimes for 4 and 5-clique finding, as a function of m [Eisenbrand and Grandoni, TCS’04]. With the current bounds on matrix multiplication, our algorithms run in O(m1.66) and O(m2.06) time, respectively, for 4-clique and 5-clique finding.Our main contribution is an output-sensitive algorithm for listing k-cliques, for any constant k≥ 3. We complement the algorithm with tight lower bounds based on standard fine-grained assumptions. Previously, the only known conditionally optimal output-sensitive algorithms were for the case of 3-cliques given by Bj'orklund, Pagh, Vassilevska W. and Zwick [ICALP’14]. If the matrix multiplication exponent ω is 2, and if the number of k-cliques t is large enough, the running time of our algorithms is Õ(min{m1/k−2t1 − 2/k(k−2),n2/k−1t1−2/k(k−1)}), and this is tight under the Exact-k-Clique Hypothesis. This running time naturally extends the running time obtained by Bj'orklund, Pagh, Vassilevska W. and Zwick for k=3.Our framework is very general in that it gives k-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<k. This generalizes the typical parameterization in terms of n (the number of 1-cliques) and m (the number of 2-cliques).If ω is 2, and if the size of the output, Δk, is sufficiently large, then for every ℓ<k, the running time of our algorithm for listing k-cliques is Õ(Δℓ2/ℓ (k − ℓ)Δk1−2/k(k−ℓ)). We also show that this runtime is optimal for all 1 ≤ ℓ < k under the Exact k-Clique hypothesis.