发表机构
Pohang University of Science and Technology (POSTECH)（韩国科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有界倍增维度度量空间中的带惩罚最小代价二分匹配问题，提出了首个近线性时间随机近似算法，能在 $O(n \mathrm{poly}(\log n, 1/\varepsilon))$ 时间内以高概率获得 $(1+\varepsilon)$-近似解。
AI 中文摘要
本文研究了有界倍增维度度量空间中带惩罚的最小代价二分匹配问题：给定度量空间 $\mathcal{M}$ 中两个不相交的集合 $R, B$，满足 $|R|+|B|=n$，以及一个惩罚函数 $p \colon R \cup B \to \mathbb{R}_{\ge 0}$，目标是选择 $R\times B$ 中的一组点对，使得每个点至多属于一个点对，并且所选点对的距离之和与不属于任何点对的点的惩罚之和最小。虽然已知在几何设置下最小代价完美匹配问题存在近线性时间近似算法，但此前在带惩罚的设置下没有这样的算法。我们提出了一种随机算法，该算法以高概率在 $O(n \mathrm{poly}(\log n, 1/\varepsilon))$ 时间内计算出 $(1+\varepsilon)$-近似的带惩罚的最小代价二分匹配。据我们所知，这是该问题在带惩罚设置下的首个近线性时间近似算法。
英文摘要
In this paper, we study the minimum-cost bipartite matching with penalties problem in metric spaces with bounded doubling dimension: Given two disjoint sets $R, B$ in a metric space $\mathcal{M}$ with $|R|+|B|=n$ and a penalty function $p \colon R \cup B \to \mathbb{R}_{\ge 0}$, the goal is to select a set of pairs in $R\times B$ so that every point belongs to at most one pair and the sum of the distances of the selected pairs and the penalties of the points not belonging to any pair is minimized. While near-linear time approximation algorithms are known for the minimum-cost perfect matching problem in geometric settings, no such algorithm was previously known for the penalty setting. We present a randomized algorithm that computes a $(1+\varepsilon)$-approximate minimum-cost bipartite matching with penalties in $O(n \mathrm{poly}(\log n, 1/\varepsilon))$ time with high probability. To the best of our knowledge, this is the first near-linear time approximation algorithm for the problem in the penalty setting.
