An $$O(n^2)$$ O ( n 2 ) Algorithm for the Limited-Capacity Many-to-Many Point Matching in One Dimension

Abstract

Given two point sets S and T , in a many-to-many matching between S and T each point in S is assigned to one or more points in T and vice versa. A generalization of the many-to-many matching problem is the limited capacity many-to-many matching problem, where the number of points that can be matched to each point (the capacity of each point) is limited. In this paper, we provide an O n 2 time algorithm for the one dimensional minimum-cost limited capacity many-to-many matching problem, where |S| + |T | = n. Our algorithm improves the best previous time complexity of O(kn 2 ), that in which k is the largest capacity of the points in S ∪ T . In this problem, both S and T lie on the real line and the cost of matching s ∈ S to t ∈ T is equal to the distance between s and t.

An O(n^2) algorithm for Many-To-Many Matching of Points with Demands in One Dimension

“…In this section, we present an O(n 2 ) algorithms for finding an OMMD between two sets S and T lying on the line. Our recursive dynamic programming algorithm is based on the algorithm of Colannino et al [4] and Rajabi and Bagheri [10]. We begin with some useful lemmas.…”

“…A special case of the LCMM problem is that in which both S and T lie on the real line. Rajabi-Alni and Bagheri [10] proposed an O(n 2 ) time algorithm for the one dimensional minimum-cost LCMM.…”

Computing a Many-to-Many Matching with Demands and Capacities between Two Sets Using the Hungarian Algorithm

“…Te minimum-cost LCMM problem can be reduced to fnding a minimum-weight DCS in an undirected bipartite graph. A one-dimensional LCMM (OLCMM), an LCMM where A and B lie on the real line, was solved in O(n 2 ) time [17]. A simple b-matching in H is a subset of the edges F ⊂ E ′ in which there is a quota b: V ⟶ Z > 0 on the vertices v ∈ V such that deg(v) ≤ b(v) for all v ∈ V. When the edge weights are integers, the optimal simple b-matching problem in the bipartite graph…”

Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane

“…Furthermore, the authors state the task of generalizing their work to this version in the plane as an important open problem, especially since geometry has helped in designing efficient algorithms for the computation of the minimum-weight perfect matching for points in the plane (see, e.g., [17,18]). Several variants of the 1-dimensional many-to-many matching problem are considered in [13,14,15].…”