There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O(Δ + log * n) communication rounds; here n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: these problems cannot be solved in o(log * n) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that maximal matchings and maximal independent sets cannot be found in o(Δ + log log n/ log log log n) rounds with any randomized algorithm in the LOCAL model of distributed computing. As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in o(Δ + log n/ log log n) rounds; this is an improvement over prior lower bounds also as a function of n. Keywords-Maximal matching; maximal independent set; distributed graph algorithms; lower bounds. • MM and MIS are strictly harder than (Δ + 1)-vertex coloring and (2Δ − 1)-edge coloring. • MM and MIS are exactly as hard as greedy coloring.
Follow-up Work In a recent breakthrough, Balliu et al. REF used our speedup technique to show that, both for maximal matching and maximal independent set, there is no randomized algorithm with runtime op∆`log log n{ log log log nq and no deterministic algorithm with runtime op∆`log n{ log log nq.
Lower Bounds for Maximal Matchings and Maximal Independent Sets
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
Computer Science