Approximation Algorithm for the Min-Cost Bipartite Matching with Penalties

2026-09-22

AI Summary

本文针对带有惩罚项的最小成本二分匹配问题，提出了一种在度量空间中运行时间接近线性的随机近似算法。

Abstract

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.

Problem

• bipartite matching
• penalties
• metric spaces
• approximation algorithm

Innovation

• approximation algorithm
• bipartite matching with penalties
• metric space
• near-linear time