Tight Distributed Listing of Cliques
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Much progress has recently been made in understanding the complexity landscape of subgraph finding problems in the CONGEST model of distributed computing. However, so far, very few tight bounds are known in this area. For triangle (i.e., 3-clique) listing, an optimal Õ(n^1/3)-round distributed algorithm has been constructed by Chang et al. [SODA 2019, PODC 2019]. Recent works of Eden et al. [DISC 2019] and of Censor-Hillel et al. [PODC 2020] have shown sublinear algorithms for K_p-listing, for each p ≥ 4, but still leaving a significant gap between the upper bounds and the known lower bounds of the problem. In this paper, we completely close this gap. We show that for each p ≥ 4, there is an Õ(n^1 - 2/p)-round distributed algorithm that lists all p-cliques K_p in the communication network. Our algorithm is optimal up to a polylogarithmic factor, due to the Ω̃(n^1 - 2/p)-round lower bound of Fischer et al. [SPAA 2018], which holds even in the CONGESTED CLIQUE model. Together with the triangle-listing algorithm by Chang et al. [SODA 2019, PODC 2019], our result thus shows that the round complexity of K_p-listing, for all p, is the same in both the CONGEST and CONGESTED CLIQUE models, at Θ̃(n^1 - 2/p) rounds. For p=4, our result additionally matches the Ω̃(n^1/2) lower bound for K_4-detection by Czumaj and Konrad [DISC 2018], implying that the round complexities for detection and listing of K_4 are equivalent in the CONGEST model.
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