发表机构
Technical University of Leoben（列文工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种运行时间为O(n log n)的新算法求解欧几里得最大权匹配问题，在测试中达到最优或接近最优解，最差差距小于1.38%，适用于大规模实例。
AI 中文摘要
在加权图$G = (V, E)$中，最大权匹配问题（MWM）要求找到其顶点的一个匹配（即配对），使得每个顶点至多与另一个顶点配对，并且所有连接配对顶点的边的权重之和最大化。如果图的顶点对应于欧几里得平面中的点，权重对应于它们之间的成对欧几里得距离，我们就得到欧几里得最大权匹配问题（Euclidean MWM）。解决该问题的最佳最优解算法运行时间为$O(n^{2.5})$。此外，存在一个FPTAS，保证$(1 - \epsilon)$近似比，运行时间为$O(m \epsilon^{-1} \log \epsilon^{-1})$。已知具有次二次运行时间（相对于顶点数$|V|$）的启发式方法，但通常产生质量一般的解。在本文中，我们提出了一种新的求解欧几里得MWM的算法，运行时间为$O(n \log n)$，并提供出色的解质量，尤其对于较大规模的实例。特别地，在我们的计算测试中，该算法为所有测试实例生成了最优或接近最优的解；观察到的最差最优性差距小于$1.38\\%$。这使得该算法在实际应用中极具吸引力，尤其是当精确方法因实例规模而计算上不可行时。
英文摘要
In a weighted graph $G = (V, E)$, the maximum weight matching problem (MWM) asks for a matching (i.e. pairing) of its vertices, such that each vertex is paired with at most one other vertex and the total sum of weights of all edges connecting paired vertices is maximised. If the vertices of the graph correspond to points in the Euclidean plane and the weights to their pairwise Euclidean distances, we get the Euclidean maximum weight matching problem (Euclidean MWM). The best optimum-solution algorithm for this problem runs in $O(n^{2.5})$. Furthermore, there exists an FPTAS guaranteeing a $(1 - ε)$-approximation ratio, which runs in $O(m ε^{-1} \log ε^{-1})$ time. Heuristics with a subquadratic running time (with respect to the number of vertices $|V|$) are known, but often yield solutions of a modest quality. In this paper, we present a novel algorithm for solving the Euclidean MWM running in $O(n \log n)$ time and providing excellent solution quality, especially for larger instances. In particular, in our computational tests, the algorithm yielded optimum or near-optimum solutions for all test instances; the worst observed optimality gap was less than $1.38\%$. This makes the algorithm highly attractive for practical applications, especially when exact methods become computationally prohibitive due to the size of the instance.
