AI 中文总结
研究直线上点集最小权重完美匹配在动态添加或删除点对时的稳定性，提出\(O(\sqrt{n})\)稳定且维持\(2 -\)近似的算法，证明\(o(\log n)\)稳定算法有无限近似比，给出全动态情形下的结果及二分变体的下界。
AI 中文摘要
计算欧几里得空间中点集\(P\)的最小权重完美匹配是一个经典几何优化问题。本文考虑动态情形，即点对可添加或从\(P\)中移除。重点是在不频繁改变解的情况下维持近似最优解。具体关注\(k -\)稳定算法，每次更新\(P\)后匹配中至多改变\(k\)条边。研究了\(\mathbb{R}^1\)中点集算法稳定性与维持解的近似比之间的权衡。首先给出一个\(O(\sqrt{n})\)稳定且维持\(2 -\)近似的算法，并证明其在所有亚线性稳定算法中最优。其次证明任何\(o(\log n)\)稳定算法具有无界近似比。下界在仅插入情形也成立，算法适用于全动态情形，且下界对问题的二分变体也成立。
英文摘要
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.
CommentsFull version of a paper that will appear in ESA 2026
