When MIS and Maximal Matching are Easy in the Congested Clique

TL;DR

This paper investigates the complexity of solving MIS and Maximal Matching in the Congested Clique model, identifying specific graph classes where these problems are efficiently solvable in constant rounds.

Contribution

The paper introduces new constant-round algorithms for MIS and Maximal Matching on graphs with certain bounded parameters, and establishes tight bounds for these parameters.

Findings

Constant-round algorithms for graphs with bounded average degree.

Constant-round algorithms for graphs with bounded neighborhood independence.

Constant-round algorithms for graphs with bounded independence number.

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 $O(\log\log \Delta)$ being the best upper bound known (where $\Delta$ 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 $\beta(G) \le 2^{O(\sqrt{\log n})}$, or (3) the independence number is at most $\alpha(G) \le |V(G)|/d(G)^{\mu}$, for any constant $\mu > 0$. 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.