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PTAS for Steiner Tree on Map Graphs

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LATIN 2020: Theoretical Informatics (LATIN 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12118))

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Abstract

We study the Steiner tree problem on map graphs, which substantially generalize planar graphs as they allow arbitrarily large cliques. We obtain a PTAS for Steiner tree on map graphs, which builds on the result for planar edge weighted instances of Borradaile et al.

The Steiner tree problem on map graphs can be casted as a special case of the planar node-weighted Steiner tree problem, for which only a 2.4-approximation is known. We prove and use a contraction decomposition theorem for planar node weighted instances. This readily reduces the problem of finding a PTAS for planar node-weighted Steiner tree to finding a spanner, \(\textit{i.e.}\), a constant-factor approximation containing a nearly optimum solution. Finally, we pin-point places where known techniques for constructing such spanner fail on node weighted instances and further progress requires new ideas.

The first three authors were supported by the NCN grant number 2015/18/E/ST6/00456.

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References

  1. Arora, S., Grigni, M., Karger, D.R., Klein, P.N., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 25–27 January 1998, San Francisco, California, USA, pp. 33–41 (1998)

    Google Scholar 

  2. Baker, B.S.: Approximation algorithms for np-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)

    Article  MathSciNet  Google Scholar 

  3. Bateni, M., Chekuri, C., Ene, A., Hajiaghayi, M.T., Korula, N., Marx, D.: Prize-collecting Steiner problems on planar graphs. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23–25, 2011, pp. 1028–1049 (2011)

    Google Scholar 

  4. Bateni, M., Demaine, E.D., Hajiaghayi, M., Marx, D.: A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, 18–21 June 2016, pp. 570–583 (2016)

    Google Scholar 

  5. Bateni, M., Farhadi, A., Hajiaghayi, M.: Polynomial-time approximation scheme for minimum k-cut in planar and minor-free graphs. In: Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, 6–9 January 2019, pp. 1055–1068 (2019)

    Google Scholar 

  6. Bateni, M., Hajiaghayi, M.T., Marx, D.: Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth. J. ACM 58(5), 21:1–21:37 (2011)

    Article  MathSciNet  Google Scholar 

  7. Berman, P., Yaroslavtsev, G.: Primal-dual approximation algorithms for node-weighted network design in planar graphs. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA, USA, 15–17 August 2012. Proceedings, pp. 50–60 (2012)

    Google Scholar 

  8. Borradaile, G., Klein, P., Mathieu, C.: An o(n log n) approximation scheme for Steiner tree in planar graphs. ACM Trans. Algorithms 5, 31:1–31:31 (2009)

    Article  Google Scholar 

  9. Buchanan, A., Wang, Y., Butenko, S.: Algorithms for node-weighted Steiner tree and maximum-weight connected subgraph. Networks 72(2), 238–248 (2018)

    Article  MathSciNet  Google Scholar 

  10. Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: Steiner tree approximation via iterative randomized rounding. J. ACM 60(1), 6:1–6:33 (2013)

    Article  MathSciNet  Google Scholar 

  11. Byrka, J., Lewandowski, M., Meesum, S.M., Spoerhase, J., Uniyal, S.: PTAS for steiner tree on map graphs. CoRR abs/1912.00717 (2019), http://arxiv.org/abs/1912.00717

  12. Cabello, S., Gajser, D.: Simple ptas’s for families of graphs excluding a minor. Discrete Appl. Math. 189, 41–48 (2015)

    Article  MathSciNet  Google Scholar 

  13. Chen, Z., Grigni, M., Papadimitriou, C.H.: Map graphs. J. ACM 49(2), 127–138 (2002)

    Article  MathSciNet  Google Scholar 

  14. Chimani, M., Mutzel, P., Zey, B.: Improved Steiner tree algorithms for bounded treewidth. J. Discrete Algorithms 16, 67–78 (2012)

    Article  MathSciNet  Google Scholar 

  15. Cohen-Addad, V., Klein, P.N., Mathieu, C.: Local search yields approximation schemes for k-means and k-median in euclidean and minor-free metrics. SIAM J. Comput. 48(2), 644–667 (2019)

    Article  MathSciNet  Google Scholar 

  16. Demaine, E.D., Hajiaghayi, M.T., Klein, P.N.: Node-weighted Steiner tree and group Steiner tree in planar graphs. ACM Trans. Algorithms 10(3), 13:1–13:20 (2014)

    Article  MathSciNet  Google Scholar 

  17. Demaine, E.D., Hajiaghayi, M.: The bidimensionality theory and its algorithmic applications. Comput. J. 51(3), 292–302 (2008)

    Article  Google Scholar 

  18. Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1(3), 195–207 (1971)

    Article  MathSciNet  Google Scholar 

  19. Eisenstat, D., Klein, P.N., Mathieu, C.: An efficient polynomial-time approximation scheme for Steiner forest in planar graphs. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, 17–19 January 2012, pp. 626–638 (2012)

    Google Scholar 

  20. Guha, S., Moss, A., Naor, J., Schieber, B.: Efficient recovery from power outage (extended abstract). In: Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, 1–4 May 1999, Atlanta, Georgia, USA, pp. 574–582 (1999)

    Google Scholar 

  21. Kammer, F., Tholey, T.: Approximate tree decompositions of planar graphs in linear time. Theoret. Comput. Sci. 645, 60–90 (2016)

    Article  MathSciNet  Google Scholar 

  22. Klein, P.N.: A subset spanner for planar graphs: with application to subset TSP. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, 21–23 May 2006, pp. 749–756 (2006)

    Google Scholar 

  23. Klein, P.N.: A linear-time approximation scheme for TSP in undirected planar graphs with edge-weights. SIAM J. Comput. 37(6), 1926–1952 (2008)

    Article  MathSciNet  Google Scholar 

  24. Klein, P.N., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19(1), 104–115 (1995)

    Article  MathSciNet  Google Scholar 

  25. Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980)

    Article  MathSciNet  Google Scholar 

  26. Moldenhauer, C.: Primal-dual approximation algorithms for node-weighted Steiner forest on planar graphs. Inf. Comput. 222, 293–306 (2013)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, University of Wrocław, Wrocław, Poland

    Jarosław Byrka, Mateusz Lewandowski & Syed Mohammad Meesum

  2. Aalto University, Espoo, Finland

    Joachim Spoerhase & Sumedha Uniyal

Authors
  1. Jarosław Byrka
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  2. Mateusz Lewandowski
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  3. Syed Mohammad Meesum
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  4. Joachim Spoerhase
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  5. Sumedha Uniyal
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Corresponding author

Correspondence to Mateusz Lewandowski .

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Editors and Affiliations

  1. University of São Paulo, São Paulo, Brazil

    Yoshiharu Kohayakawa

  2. University of Campinas, Campinas, Brazil

    Flávio Keidi Miyazawa

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Byrka, J., Lewandowski, M., Meesum, S.M., Spoerhase, J., Uniyal, S. (2020). PTAS for Steiner Tree on Map Graphs. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9_1

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  • DOI: https://doi.org/10.1007/978-3-030-61792-9_1

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  • Print ISBN: 978-3-030-61791-2

  • Online ISBN: 978-3-030-61792-9

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