Dynamic Approximate Matchings with an Optimal Recourse Bound

In the area of dynamic matching algorithms, the number of changes to the matching per update step, or the {recourse bound}, is an important measure of quality. Nevertheless, the worst-case recourse bounds of almost all known dynamic approximate matching algorithms are prohibitively large, significantly larger than the corresponding update times. In this paper we fill in this gap via a surprisingly simple observation: Any dynamic algorithm for maintaining a $\beta$-approximate maximum matching with update time $T$, for any $\beta \ge 1$, $T$ and $\epsilon > 0$, can be transformed into an algorithm for maintaining a $(\beta(1 +\epsilon))$-approximate maximum matching with update time $T + O(1/\epsilon)$ and recourse bound $O(1/\epsilon)$. If the original update time $T$ is amortized/worst-case, so is the resulting update time $T + O(1/\epsilon)$, while the recourse bound $O(1/\epsilon)$ holds in the worst-case. This transformation applies to the fully dynamic setting under edge updates and vertex updates. We complement this positive result by showing that, for $\beta = 1+\epsilon$, the worst-case recourse bound provided by our transformation is optimal: For any $\epsilon = \Omega(1/n)$, maintaining a $(1+\epsilon)$-approximate maximum matching in an $n$-vertex graph requires $\Omega(1/\epsilon)$ changes to the matching per step, even in the amortized sense and even in the incremental and decremental settings. As a corollary, several key dynamic approximate matching algorithms, with low update time bounds but poor recourse bounds, are strengthened to achieve near-optimal recourse bounds with no loss in the update time. Furthermore, although a direct application of this transformation may only hurt the update time, we demonstrate the usefulness of this transformation in achieving low update time bounds in some natural settings (where we might not care about recourse bounds).