Dynamic Approximate Maximum Matching in the Distributed Vertex Partition Model

Peter Robinson, Xianbin Zhu

Abstract: We initiate the study of approximate maximum matching in the vertex partition model, for graphs subject to dynamic changes. We assume that the $n$ vertices of the graph are partitioned among $k$ players, who execute a distributed algorithm and communicate via message passing. An adaptive adversary may perform dynamic updates to the graph topology by inserting or removing edges between the nodes, and the algorithm needs to respond to these changes by adapting the output of the players, with the goal of maintaining an approximate maximum matching. The main performance metric in this setting is the algorithm's update time, which corresponds to the number of rounds required for updating the solution upon an adversarial change. For the standard setting of single-edge insertions and deletions, we obtain the following results: We give a randomized Las Vegas algorithm with an expected update time of $O( \frac{\sqrt{m}}{\beta k} )$ rounds that maintains a $\frac{2}{3}$-approximate maximum matching that is also maximal, where $m$ is the number of edges of the graph. We also show that any algorithm has a worst case update time of $\Omega( \frac{n}{\beta k^2\log n} )$, assuming a link bandwidth of $O(\beta\log n)$ bits per round, if it maintains a matching that is maximal and does not have any 3-augmenting paths. For batch-dynamic updates, where the adversary may modify up to $\ell\ge 1$ edges at once, we prove the following: There is a randomized algorithm that succeeds with high probability in maintaining a $\frac{2}{3}$-approximate maximum matching and has a worst case update time of $\Omega( \frac{\ell\log n}{\sqrt{\beta k}} )$ rounds. We show that $\Omega( \frac{\ell}{\beta k \log n} )$ poses a lower bound for maintaining a maximal matching without 3-augmenting paths.