Matching problems comprise an important set of problems that link the areas of graph theory and combinatorial optimization. The maximum cardinality matching problem (see below) is one of the first integer programming problems that was solved in polynomial time. Matchings are of great importance in graph theory (see [9]) as well as in combinatorial optimization (see e.g. [15]).
The matching problem and its variations arise in cases when we want to find an ‘optimal’ pairing of the members of two (not necessarily disjoint) sets. In particular, if we are given two sets of ‘objects’ and a ‘weight’ for each pair of objects, we want to match the objects into pairs in such a way that the total weight is maximal. In graph theory, the problem is defined on a graph G = (V, E) where V is the node set of the graph, corresponding to the union of the two sets of objects, and Eis the edge set of the graph corresponding to the possible pairs. A pair is possible if there exists an edge between the...
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Alevras, D. (2001). Assignment and Matching . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_12
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