Faster sublinear approximation of the number of k-cliques in low-arboricity graphs

Given query access to an undirected graph G, we consider the problem of computing a (1 ± ε)-approximation of the number of k-cliques in G. The standard query model for general graphs allows for degree queries, neighbor queries, and pair queries. Let n be the number of vertices, m be the number of edges, and n_k be the number of k-cliques. Previous work by Eden, Ron and Seshadhri (STOC 2018) n gives an O^∗(_n1/k + ^mk/2 )-time algorithm for this problem nk k (we use O^∗(·) to suppress poly(log n, 1/ε, k^k) dependencies). Moreover, this bound is nearly optimal when the expression is sublinear in the size of the graph. Our motivation is to circumvent this lower bound, by parameterizing the complexity in terms of graph arboricity. The arboricity of G is a measure for the graph density “everywhere”. There is a very rich family of graphs with bounded arboricity, including all minor-closed graph classes (such as planar graphs and graphs with bounded treewidth), bounded degree graphs, preferential attachment graphs and more. We design an algorithm for the class of graphs with arboricity at most α, whose running time is + ^mαk−2 n O^∗(min{^nα_n^k_k⁻1 , }). We also prove a nearly n¹/k nk k matching lower bound. For all graphs, the arboricity is O(√m), so this bound subsumes all previous results on sublinear clique approximation. As a special case of interest, consider minor-closed families of graphs, which have constant arboricity. Our result implies that for any minor-closed family of graphs, there is a (1 ± ε)-approximation algorithm for n_k that has running time O^∗(_nⁿ_k ). Such a bound was not known even for the special (classic) case of triangle counting in planar graphs.