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/ωmin, n/ε})O(1/ε). (Δ O(1/ε) + log*(n)) rounds for edge-weights in [wmin, 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 wmin 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/wmin are sufficiently slowly increasing functions of n. Moreover, our algorithms are deterministic rather than randomized.