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 $(1\eps)$ times the optimal in$\Delta^{O(1/\eps)} + O\left(\frac{1}{\eps^2}\right) \cdot\log^*(n)$ rounds where $n$ is the number of vertices in the graph and $\Delta$ is the maximum degree. Our algorithm for the edgeweighted case computes a matching whose weight is at least $(1\eps)$ times the optimal in$\log(\min\{1/\wmin,n/\eps\})^{O(1/\eps)}\cdot(\Delta^{O(1/\eps)}+\log^*(n))$ rounds for edgeweights in $[\wmin,1]$.
The best previous algorithms for both the unweighted case and the weighted case are by Lotker, PattShamir, and Pettie~(SPAA 2008). For the unweighted case they give a randomized $(1\eps)$approximation algorithm that runs in $O((\log(n)) /\eps^3)$ rounds. For the weighted case they give a randomized $(1/2\eps)$approximation algorithm that runs in $O(\log(\eps^{1}) \cdot \log(n))$ rounds. Hence, our results improve on the previous ones when the parameters Δ, $\eps$ and $\wmin$ are constants (where we reduce the number of runs from O(log(n)) to O(log∗(n))), and more generally when Δ, $1/\eps$ and $1/\wmin$ are sufficiently slowly increasing functions of n. Moreover, our algorithms are deterministic rather than randomized.
