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Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

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Title: Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
Authors: Klemz, Boris, Rote, Günter
Publication Year: 2022
Collection: FU Berlin: Refubium
Subject Terms: Graph algorithm, Induced matching, Chain cover, Convex bipartite graph, Certifying algorithm, Dynamic programming, ddc:004
Description: A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex u∈U, the neighbors of u are consecutive in the ordering of V. An induced matching H of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex u∈U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n2) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.
Document Type: article in journal/newspaper
File Description: 17 Seiten; application/pdf
Language: English
DOI: 10.17169/refubium-33566
DOI: 10.1007/s00453-021-00904-w
Availability: https://refubium.fu-berlin.de/handle/fub188/33847
https://doi.org/10.17169/refubium-33566
https://doi.org/10.1007/s00453-021-00904-w
Rights: https://creativecommons.org/licenses/by/4.0/
Accession Number: edsbas.D82F7DC5
Database: BASE
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  – Url: https://refubium.fu-berlin.de/handle/fub188/33847#
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  Label: Title
  Group: Ti
  Data: Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
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  Data: <searchLink fieldCode="AR" term="%22Klemz%2C+Boris%22">Klemz, Boris</searchLink><br /><searchLink fieldCode="AR" term="%22Rote%2C+Günter%22">Rote, Günter</searchLink>
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  Data: 2022
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  Data: FU Berlin: Refubium
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  Data: <searchLink fieldCode="DE" term="%22Graph+algorithm%22">Graph algorithm</searchLink><br /><searchLink fieldCode="DE" term="%22Induced+matching%22">Induced matching</searchLink><br /><searchLink fieldCode="DE" term="%22Chain+cover%22">Chain cover</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+bipartite+graph%22">Convex bipartite graph</searchLink><br /><searchLink fieldCode="DE" term="%22Certifying+algorithm%22">Certifying algorithm</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamic+programming%22">Dynamic programming</searchLink><br /><searchLink fieldCode="DE" term="%22ddc%3A004%22">ddc:004</searchLink>
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  Data: A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex u∈U, the neighbors of u are consecutive in the ordering of V. An induced matching H of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex u∈U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n2) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.
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      – Text: English
    Subjects:
      – SubjectFull: Graph algorithm
        Type: general
      – SubjectFull: Induced matching
        Type: general
      – SubjectFull: Chain cover
        Type: general
      – SubjectFull: Convex bipartite graph
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      – SubjectFull: Certifying algorithm
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      – SubjectFull: Dynamic programming
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      – TitleFull: Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
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            NameFull: Klemz, Boris
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              Y: 2022
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