Lower Bounds for Maximal Matchings and Maximal Independent Sets

There are distributed graph algorithms for finding maximal matchings and independent sets in O ( Δ + log * n ) communication rounds; here, is the number of nodes maximum degree. The lower bound by Linial (1987, 1992) shows that dependency on optimal: These problems cannot be solved o (log rounds even if = 2. However, a long-standing open question, there currently an exponential gap between upper bounds. We prove bounds tight. show any algorithm finds matching or set with probability at least 1-1/ requires Ω (min { , / }) LOCAL model computing. As corollary, it follows deterministic this improvement over prior also as function .