Low-Sensitivity Matching via Sampling from Gibbs Distributions
Yuichi Yoshida, Zihan Zhang
TL;DR
The paper tackles the sensitivity of the maximum matching problem under edge deletions by leveraging sampling from a Gibbs distribution over matchings. It develops a low-sensitivity, near-optimal algorithm that achieves a $(1-\varepsilon)$-approximation with sensitivity roughly $\Delta^{O(1/\varepsilon)}$ for bounded-degree graphs, using a carefully chosen Gibbs parameter $\lambda$ and recursive coupling analyses. For planar and bipartite graphs, it introduces faster sampling-based schemes via a 2-site Glauber dynamic, yielding substantial runtime improvements. To handle general graphs, it employs entropy-regularized LP-based sparsification to reduce to a low-degree instance, preserving approximate optimum and enabling the prior algorithm with comparable sensitivity bounds, while establishing a lower bound that forces exponential dependence on $1/\varepsilon$ for the Gibbs parameter. Overall, the results substantially advance the trade-off between approximation quality and sensitivity, with implications for trustworthy decision-making and privacy-aware data analysis in graph optimization.
Abstract
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 $Δ^{O(1/\varepsilon)}$, where $Δ$ is the maximum degree of the input graph. The algorithm is based on sampling from the Gibbs distribution over matchings and runs in time $O_{\varepsilon, Δ}(m \log m)$, 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 $Δ$, 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 $O(n^{1/(1+\varepsilon^2)})$.