Abstract
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. We seek for a maximum set of edges, that 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 subset’s bound of edges, and so on hierarchically). This is a sub-problem of the matroid matching problem which is \(\textsc {NP}\text {-hard}\) in general. It corresponds to the special case where the matroid is restricted to be laminar and the weights are unity. A pseudo-polynomial algorithm for the weighted laminar matroid matching problem is presented in [8]. We propose a polynomial-time algorithm for Hierarchical b-matching, i.e. the unweighted laminar matroid matching problem, and discuss how our techniques can possibly be generalized to the weighted case.
This research was supported in part by PetaCloud - a project funded by the Israel Innovation Authority.
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Emek, Y., Kutten, S., Shalom, M., Zaks, S. (2021). Hierarchical b-Matching. In: Bureš, T., et al. SOFSEM 2021: Theory and Practice of Computer Science. SOFSEM 2021. Lecture Notes in Computer Science(), vol 12607. Springer, Cham. https://doi.org/10.1007/978-3-030-67731-2_14
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DOI: https://doi.org/10.1007/978-3-030-67731-2_14
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