TY - GEN
T1 - Hierarchical b-Matching
AU - Emek, Yuval
AU - Kutten, Shay
AU - Shalom, Mordechai
AU - Zaks, Shmuel
N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - 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 bv, and a maximum b-matching is a maximum set of edges, such that every vertex v appears in at most bv 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 bv 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 NP -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.
AB - 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 bv, and a maximum b-matching is a maximum set of edges, such that every vertex v appears in at most bv 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 bv 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 NP -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.
KW - Matching
KW - Matroids
KW - b-matching
UR - http://www.scopus.com/inward/record.url?scp=85101552365&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-67731-2_14
DO - 10.1007/978-3-030-67731-2_14
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AN - SCOPUS:85101552365
SN - 9783030677305
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 189
EP - 202
BT - SOFSEM 2021
A2 - Bureš, Tomáš
A2 - Dondi, Riccardo
A2 - Gamper, Johann
A2 - Guerrini, Giovanna
A2 - Jurdzinski, Tomasz
A2 - Pahl, Claus
A2 - Sikora, Florian
A2 - Wong, Prudence W.
T2 - 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021
Y2 - 25 January 2021 through 29 January 2021
ER -