Lower Bounds for Maximal Matchings and Maximal Independent Sets

AbstractThere 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 any algorithm that finds a maximal matching or maximal independent set with probability at least 1-1/n requires Ω (min { Δ, log log n / log log log n}) rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires Ω (min {Δ, log n / log log n}) rounds; this is an improvement over prior lower bounds also as a function of n.