A Faster Algorithm for the Limited-Capacity Many-to-Many Point Matching in One Dimension

Given two point sets S and T on a line, we present the first linear time algorithm for finding the limited capacity many-to-many matching (LCMM) between S and T improving the previous best known quadratic time algorithm. The aim of the LCMM is to match each point of S (T) to at least one point of T (S) such that the matching costs is minimized and the number of the points matched to each point is limited to a given number.Comment: 18 pages, 7 figures. arXiv admin note: text overlap with arXiv:1702.0108

A Fast and Efficient algorithm for Many-To-Many Matching of Points with Demands in One Dimension

Given two point sets S and T, a 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.Comment: 14 pages,8 figures. arXiv admin note: substantial text overlap with arXiv:1702.0108

Computing a many-to-many matching with demands and capacities between two sets using the Hungarian algorithm

Given two sets A={a_1,a_2,...,a_s} and {b_1,b_2,...,b_t}, a many-to-many matching with demands and capacities (MMDC) between A and B matches each element a_i in A to at least \alpha_i and at most \alpha'_i elements in B, and each element b_j in B to at least \beta_j and at most \beta'_j elements in A for all 1=<i<=s and 1=<j<=t. In this paper, we present an algorithm for finding a minimum-cost MMDC between A and B using the well-known Hungarian algorithm.Comment: 8 pages, 1 figur