On the Stability of Minimum-Weight Perfect Matching on the Line

2026-07-17

AI Summary

This work addresses the dynamic minimum-weight perfect matching problem in one-dimensional Euclidean space, where the point set undergoes insertions and deletions, and the goal is to maintain an approximately optimal matching while bounding the number of edge changes per update—referred to as stability. The paper presents the first fully dynamic algorithm achieving both a 2-approximation ratio and $O(\sqrt{n})$ stability. Furthermore, it establishes a fundamental trade-off between stability and approximation quality by proving that any algorithm with $o(\log n)$ stability necessarily incurs an unbounded approximation ratio. This lower bound holds even in the insertion-only setting and extends to the bipartite variant of the problem. The approach integrates techniques from online algorithm design, constrained backtracking mechanisms, and adversarial lower-bound analysis.

Abstract

Computing a minimum-weight perfect matching for a point set $P$ in Euclidean space is a classic geometric optimization problem. We consider the problem in a dynamic setting, where pairs of points may be added to or removed from the set $P$. Our focus is on maintaining an approximately optimal solution without making too many changes to the solution. More precisely, we are interested in $k$-stable algorithms, which change at most $k$ edges in the matching after each update to the set $P$. In other words, we consider an online setting (with insertions and deletions) with bounded recourse. We study trade-offs between the stability of the algorithm and the approximation ratio of the maintained solution for point sets in $\mathbb{R}^1$. First, we present an $O(\sqrt{n})$-stable algorithm that maintains a $2$-approximation, which we show to be optimal among all algorithms with sublinear stability. Second, we prove that any $o(\log n)$-stable algorithm has unbounded approximation ratio. Our lower bounds hold even in the insertion-only case, while our algorithm works in the fully dynamic case. Moreover, our lower bounds also hold for the bipartite variant of the problem.

Problem

• dynamic matching
• minimum-weight perfect matching
• stability
• approximation ratio
• bounded recourse

Innovation

• dynamic matching
• bounded recourse
• stability
• approximation ratio
• online algorithm