Limited-Capacity Many-To-Many Point Matching in One Dimension

TL;DR

This paper presents an efficient O(n^2) algorithm for solving the one-dimensional minimum-cost limited capacity many-to-many matching problem, improving upon previous algorithms by reducing complexity.

Contribution

It introduces a novel O(n^2) time algorithm for limited capacity matching on a line, surpassing prior methods with higher complexity.

Findings

Algorithm runs in O(n^2) time

Improves previous complexity bounds

Applicable to real-line point sets

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, that is 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(k(n^2)), that in which k is the largest capacity of the points in the union of S and T. In this problem, both S and T lie on the real line and the cost of matching s in S to t in T is equal to the distance between s and t.