Distributed maximum matching in bounded degree graphs

We present deterministic distributed algorithms for computing approximate maximum cardinality matchings and approximate maximum weight matchings. Our algorithm for the unweighted case computes a matching whose size is at least (1 - ε) times the optimal in Δ^O(1/ε) + O (1/ε²)·log∗(n) rounds where n is the number of vertices in the graph and Δ is the maximum degree. Our algorithm for the edge-weighted case computes a matching whose weight is at least (1 - ε) times the optimal in log(min{1/w_min, n/ε})^O(1/ε)·(Δ^O(1/ε) + log∗(n)) rounds for edge-weights in [w_min, 1]. The best previous algorithms for both the unweighted case and the weighted case are by Lotker, Patt-Shamir, and Pettie (SPAA 2008). For the unweighted case they give a randomized (1 - ε)-approximation algorithm that runs in O((log(n))/ε³) rounds. For the weighted case they give a randomized (1/2 - ε)-approximation algorithm that runs in O(log(ε^-1)·log(n)) rounds. Hence, our results improve on the previous ones when the parameters Δ, ε and w_min are constants (where we reduce the number of runs from O(log(n)) to O(log∗(n))), and more generally when Δ, 1/ε and 1/w_min are sufficiently slowly increasing functions of n. Moreover, our algorithms are deterministic rather than randomized.