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Approximations for the Steiner Multicycle Problem

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LATIN 2022: Theoretical Informatics (LATIN 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13568))

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Abstract

The Steiner Multicycle problem consists in, given a complete graph G, a weight function \(w :E(G) \rightarrow \mathbb {Q}_+\), and a partition of V(G) into terminal sets, finding a minimum-weight collection of disjoint cycles in G such that, for every terminal set T, all vertices of T are in a same cycle of the collection. This problem, which is motivated by applications on routing problems with pickup and delivery locations, generalizes the Traveling Salesman problem (TSP) and therefore is hard to approximate in general. Using an algorithm for the Survivable Network Design problem and T-joins, we obtain a 3-approximation for its metric case, improving on the previous best 4-approximation. Furthermore, inspired by a result by Papadimitriou and Yannakakis for the \(\{1,2\}\)-TSP, we present an (11/9)-approximation for the particular case of the Steiner Multicycle in which each edge weight is 1 or 2. This algorithm can be adapted into a (7/6)-approximation when every terminal set contains at least 4 vertices.

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Acknowledgement

C. G. Fernandes was partially supported by the National Council for Scientific and Technological Development – CNPq (Proc. 310979/2020-0 and 423833/2018-9). C. N. Lintzmayer was partially supported by CNPq (Proc. 312026/2021-8 and 428385/2018-4). P. F. S. Moura was partially supported by the Fundação de Amparo à Pesquisa do Estado de Minas Gerais – FAPEMIG (APQ-01040-21). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, and by Grant #2019/13364-7, São Paulo Research Foundation (FAPESP).

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Authors and Affiliations

  1. Department of Computer Science, University of São Paulo, São Paulo, Brazil

    Cristina G. Fernandes

  2. Center for Mathematics, Computing and Cognition, Federal University of ABC, Santo André, São Paulo, Brazil

    Carla N. Lintzmayer

  3. Computer Science Department, Federal University of Minas Gerais, Belo Horizonte, Minas Gerais, Brazil

    Phablo F. S. Moura

Authors
  1. Cristina G. Fernandes
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  2. Carla N. Lintzmayer
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  3. Phablo F. S. Moura
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Corresponding author

Correspondence to Carla N. Lintzmayer .

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Editors and Affiliations

  1. Universidad Nacional Autónoma de México, Mexico City, Mexico

    Armando Castañeda

  2. Centro de investigación y de Estudios Avanzados, Mexico City, Mexico

    Francisco Rodríguez-Henríquez

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Fernandes, C.G., Lintzmayer, C.N., Moura, P.F.S. (2022). Approximations for the Steiner Multicycle Problem. In: Castañeda, A., Rodríguez-Henríquez, F. (eds) LATIN 2022: Theoretical Informatics. LATIN 2022. Lecture Notes in Computer Science, vol 13568. Springer, Cham. https://doi.org/10.1007/978-3-031-20624-5_12

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  • DOI: https://doi.org/10.1007/978-3-031-20624-5_12

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  • Online ISBN: 978-3-031-20624-5

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