Given two point sets S and T, we first study the many-to-many matching with demands (MMD) problem , where each point of one set must be matched to at least a given number of the points of the other set . We propose an O(n^2) time algorithm for computing a one dimensional MMD (OMMD) of minimum cost  , 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 is equal to 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 an O(n^2) time algorithm for the minimum-cost one dimensional MMDC (OMMDC) problem.
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.
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