Given two point sets S and T, the minimum-cost many-to-many matching with demands (MMD) problem is the problem of finding a minimum-cost many-to-many matching between S and T such that each point of S (respectively T) is matched to at least a given number of the points of T (respectively S). We propose the first O(n^2) time algorithm for computing a one dimensional MMD (OMMD) of minimum cost between S and T, where |S|+|T|=n. In an OMMD problem, the input point sets S and T lie on the real line and the cost of matching a point to another point equals the distance between the two points. We also study a generalized version of the MMD problem, the many-to-many matching with demands and capacities (MMDC) problem, that in which each point has a limited capacity in addition to a demand. We give the first O(n^2) time algorithm for the minimum-cost one dimensional MMDC (OMMDC) problem.

翻译：给定两个点集S和T，最小成本多对多带需求匹配（MMD）问题旨在寻找S与T之间的最小成本多对多匹配，使得S（或T）中的每个点至少与T（或S）中给定数量的点相匹配。本文提出了首个计算S与T之间最小成本一维MMD（OMMD）的O(n^2)时间算法，其中|S|+|T|=n。在一维MMD问题中，输入点集S和T位于实轴上，两点匹配的成本等于它们之间的距离。进一步，我们研究了MMD问题的推广形式，即带需求与容量约束的多对多匹配（MMDC）问题，该问题中每个点除了需求外还有容量上限。我们给出了首个针对最小成本一维MMDC（OMMDC）问题的O(n^2)时间算法。