Low-Sensitivity Matching via Sampling from Gibbs Distributions

In this work, we study the maximum matching problem from the perspective of sensitivity. The sensitivity of an algorithm $A$ on a graph $G$ is defined as the maximum Wasserstein distance between the output distributions of $A$ on $G$ and on $G - e$, where $G - e$ is the graph obtained by deleting an edge $e$ from $G$. The maximum is taken over all edges $e$, and the underlying metric for the Wasserstein distance is the Hamming distance. We first show that for any $\varepsilon > 0$, there exists a polynomial-time $(1 - \varepsilon)$-approximation algorithm with sensitivity $\Delta^{O(1/\varepsilon)}$, where $\Delta$ is the maximum degree of the input graph. The algorithm is based on sampling from the Gibbs distribution over matchings and runs in time $O_{\varepsilon, \Delta}(m \log m)$, where $m$ is the number of edges in the graph. This result significantly improves the previously known sensitivity bounds. Next, we present significantly faster algorithms for planar and bipartite graphs as a function of $\varepsilon$ and $\Delta$, which run in time $\mathrm{poly}(n/\varepsilon)$. This improvement is achieved by designing a more efficient algorithm for sampling matchings from the Gibbs distribution in these graph classes, which improves upon the previous best in terms of running time. Finally, for general graphs with potentially unbounded maximum degree, we show that there exists a polynomial-time $(1 - \varepsilon)$-approximation algorithm with sensitivity $\sqrt{n} \cdot (\varepsilon^{-1} \log n)^{O(1/\varepsilon)}$, improving upon the previous best bound of $O(n^{1/(1+\varepsilon^2)})$.