  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 edge-weighted 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 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 $(1-\eps)$-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 $\Delta$, $\eps$ and $\wmin$ are constants (where we reduce the
number of runs from $O(\log(n))$ to $O(\log^*(n))$), and more generally when
$\Delta$, $1/\eps$ and $1/\wmin$ are sufficiently slowly increasing functions
of $n$. Moreover, our algorithms are deterministic rather than randomized.
