Tight Distributed Listing of Cliques

TL;DR

This paper establishes tight bounds for distributed algorithms listing all cliques of size p in a network, showing the optimal round complexity for all p ≥ 4 and unifying the complexity for detection and listing of K4.

Contribution

It provides the first tight, optimal distributed algorithms for listing all p-cliques for all p ≥ 4, closing the gap between upper and lower bounds.

Findings

Optimal or p-clique listing in distributed networks

Round complexity matches lower bounds up to polylogarithmic factors

Detection and listing complexities are shown to be equivalent for K4

Abstract

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 $\tilde{O}(n^{1/3})$-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 $p \geq 4$, 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 $p \geq 4$, there is an $\tilde{O}(n^{1 - 2/p})$-round distributed algorithm that lists all $p$-cliques $K_p$ in the communication network. Our algorithm is \emph{optimal} up to a polylogarithmic factor, due to the $\tilde{\Omega}(n^{1 - 2/p})$-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 $\tilde{\Theta}(n^{1 - 2/p})$ rounds. For $p=4$, our result additionally matches the $\tilde{\Omega}(n^{1/2})$ lower bound for $K_4$-\emph{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.