When MIS and Maximal Matching are Easy in the Congested Clique
Keren Censor-Hillel, Tomer Even, Maxime Flin, Magnús M. Halldórsson
TL;DR
The paper characterizes when MIS and MM can be solved in constant rounds in the Congested Clique, tying feasibility to graph parameters such as average degree $d(G)$, neighborhood independence $\beta(G)$, and independence number $\alpha(G)$. It introduces average-degree–driven sparsification and a non-uniform sampling scheme across degree classes to push the residual maximum degree down, enabling constant-round MIS/MM under specific bounds, and provides tight analyses showing these bounds are tight for current methods. A key ingredient is opportunistic routing to learn $r$-hop neighborhoods, enabling LOCAL-style techniques to be simulated in the Congested Clique. The paper also constructs hard graph distributions to prove that, beyond these parameter regimes, existing approaches require at least $\Omega(\log\log n)$ rounds, underscoring the need for new ideas for broader classes. Overall, the work clarifies the boundary between fast and slow MIS/MM in the Congested Clique and offers practical algorithms for graph families common in applications.
Abstract
Two of the most fundamental distributed symmetry-breaking problems are that of finding a maximal independent set (MIS) and a maximal matching (MM) in a graph. It is a major open question whether these problems can be solved in constant rounds of the all-to-all communication model of \textsf{Congested\ Clique}, with $O(\log\log Δ)$ being the best upper bound known (where $Δ$ is the maximum degree). We explore in this paper the boundary of the feasible, asking for \emph{which graphs} we can solve the problems in constant rounds. We find that for several graph parameters, ranging from sparse to highly dense graphs, the problems do have a constant-round solution. In particular, we give algorithms that run in constant rounds when: (1) the average degree is at most $d(G) \le 2^{O(\sqrt{\log n})}$, (2) the neighborhood independence number is at most $β(G) \le 2^{O(\sqrt{\log n})}$, or (3) the independence number is at most $α(G) \le |V(G)|/d(G)^μ$, for any constant $μ> 0$. Further, we establish that these are tight bounds for the known methods, for all three parameters, suggesting that new ideas are needed for further progress.