Hierarchical b-Matching
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A matching of a graph is a subset of edges no two of which share a common vertex, and a maximum matching is a matching of maximum cardinality. In a b-matching every vertex v has an associated bound b_v, and a maximum b-matching is a maximum set of edges, such that every vertex v appears in at most b_v of them. We study an extension of this problem, termed Hierarchical b-Matching. In this extension, the vertices are arranged in a hierarchical manner. At the first level the vertices are partitioned into disjoint subsets, with a given bound for each subset. At the second level the set of these subsets is again partitioned into disjoint subsets, with a given bound for each subset, and so on. In an Hierarchical b-matching we look for a maximum set of edges, that will obey all bounds (that is, no vertex v participates in more than b_v edges, then all the vertices in one subset do not participate in more that that subset's bound of edges, and so on hierarchically). We propose a polynomial-time algorithm for this new problem, that works for any number of levels of this hierarchical structure.
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