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.

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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.

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Background: A matching between two sets A and B assigns some elements of A to some elements of B. Finding the similarity between two sets of elements by advantage of the matching is widely used in computational biology for example in the contexts of genome-wide and sequencing association studies. Frequently, the capacities of the elements are limited. That is, the number of the elements that can be matched to each element should not exceed a given number. Results:We use bipartite graphs to model relationships between pairs of objects. Given an undirected bipartite graph G = (A ∪ B, E), the b-matching of G matches each vertex v in A (resp. B) to at least 1 and at most b(v) vertices in B (resp. A), where b(v) denotes the capacity of v. We propose the first O(n 3 ) time algorithm for finding the maximum weight b-matching of G, where |A| + |B| = O(n). Conclusions: The b-matching has been studied widely for the bipartite graphs with integer weight edges. But our algorithm is the first algorithm for the maximum (respectively minimum) b-matching problem with non positive real (respectively non negative real) edge weights.