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Integer Programming Formulations and Cutting Plane Algorithms for the Maximum Selective Tree Problem

Authors Ömer Burak Onar, Tınaz Ekim, Z. Caner Taşkın

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Author Details

Ömer Burak Onar
  • Department of Industrial Engineering, Bogazici University, Turkey
Tınaz Ekim
  • Department of Industrial Engineering, Bogazici University, Turkey
Z. Caner Taşkın
  • Department of Industrial Engineering, Bogazici University, Turkey

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Ömer Burak Onar, Tınaz Ekim, and Z. Caner Taşkın. Integer Programming Formulations and Cutting Plane Algorithms for the Maximum Selective Tree Problem. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SEA.2023.13

Abstract

This paper considers the Maximum Selective Tree Problem (MSelTP) as a generalization of the Maximum Induced Tree problem. Given an undirected graph with a partition of its vertex set into clusters, MSelTP aims to choose the maximum number of vertices such that at most one vertex per cluster is selected and the graph induced by the selected vertices is a tree. To the best of our knowledge, MSelTP has not been studied before although several related optimization problems have been investigated in the literature. We propose two mixed integer programming formulations for MSelTP; one based on connectivity constraints, the other based on cycle elimination constraints. In addition, we develop two exact cutting plane procedures to solve the problem to optimality. On graphs with up to 25 clusters, up to 250 vertices, and varying densities, we conduct computational experiments to compare the results of two solution procedures with solving a compact integer programming formulation of MSelTP. Our experiments indicate that the algorithm CPAXnY outperforms the other procedures overall except for graphs with low density and large cluster size, and that the algorithm CPAX yields better results in terms of the average time of instances optimally solved and the overall average time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Integer programming
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Network optimization
Keywords
  • maximum induced tree
  • selective tree
  • cutting plane
  • separation algorithm
  • mixed integer programming

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References

  1. John Adrian Bondy and Uppaluri Siva Ramachandra Murty. Graph Theory with Applications. Elsevier Science Publishing Co., Inc., 1976. Google Scholar
  2. M. Chudnovsky and Paul D. Seymour. The three-in-a-tree problem. Combinatorica, 30:387-417, 2010. Google Scholar
  3. Ovidiu Cosma, Petrică C. Pop, and Laura Cosma. An effective hybrid genetic algorithm for solving the generalized traveling salesman problem. In Hugo Sanjurjo González, Iker Pastor López, Pablo García Bringas, Héctor Quintián, and Emilio Corchado, editors, Hybrid Artificial Intelligent Systems, pages 113-123, Cham, 2021. Springer International Publishing. Google Scholar
  4. Ernando Gomes de Sousa, Rafael Castro de Andrade, and Andréa C. Santos. A multigraph formulation for the generalized minimum spanning tree problem. In ISCO 2018: Combinatorial Optimization, volume 10856 of Lecture Notes in Computer Science, pages 133-143. Springer, July 2018. URL: https://doi.org/10.1007/978-3-319-96151-4_12.
  5. Marc Demange, Tınaz Ekim, and Bernard Ries. On the minimum and maximum selective graph coloring problems in some graph classes. Discrete Applied Mathematics, 204:77-89, 2016. URL: https://doi.org/10.1016/j.dam.2015.10.005.
  6. Marc Demange, Tınaz Ekim, Bernard Ries, and Cerasela Tanasescu. On some applications of the selective graph coloring problem. European Journal of Operational Research, 240(2):307-314, 2015. URL: https://doi.org/10.1016/j.ejor.2014.05.011.
  7. Marc Demange, Jérôme Monnot, Petrica Pop, and Bernard Ries. On the complexity of the selective graph coloring problem in some special classes of graphs. Theoretical Computer Science, 540-541:89-102, 2014. Combinatorial Optimization: Theory of algorithms and Complexity. URL: https://doi.org/10.1016/j.tcs.2013.04.018.
  8. Nicolas Derhy and Christophe Picouleau. Finding induced trees. Discrete Applied Mathematics, 157:3552-3557, October 2009. URL: https://doi.org/10.1016/j.dam.2009.02.009.
  9. C.W Duin, A Volgenant, and S Voß. Solving group Steiner problems as Steiner problems. European Journal of Operational Research, 154(1):323-329, 2004. URL: https://doi.org/10.1016/S0377-2217(02)00707-5.
  10. Paul Erdös and Zbigniew Palka. Trees in random graphs. Discret. Math., 46:145-150, 1983. Google Scholar
  11. Paul Erdös, Michael Saks, and Vera T Sós. Maximum induced trees in graphs. Journal of Combinatorial Theory, Series B, 41(1):61-79, 1986. URL: https://doi.org/10.1016/0095-8956(86)90028-6.
  12. Corinne Feremans, Martine Labbé, and Gilbert Laporte. A comparative analysis of several formulations for the generalized minimum spanning tree problem. Networks, 39:29-34, January 2002. URL: https://doi.org/10.1002/net.10009.
  13. Corinne Feremans, Martine Labbé, and Gilbert Laporte. Generalized network design problems. European Journal of Operational Research, 148(1):1-13, 2003. URL: https://doi.org/10.1016/S0377-2217(02)00404-6.
  14. Corinne Feremans, Martine Labbé, Gilbert Laporte, and Etudes Commerciales. The generalized minimum spanning tree problem: Polyhedral analysis and branch-and-cut algorithm. Networks, 43, November 2002. URL: https://doi.org/10.1002/net.10105.
  15. Cristiane S. Ferreira, Luis Satoru Ochi, Víctor Parada, and Eduardo Uchoa. A GRASP-based approach to the generalized minimum spanning tree problem. Expert Systems with Applications, 39(3):3526-3536, 2012. URL: https://doi.org/10.1016/j.eswa.2011.09.043.
  16. Yuri Frota, Nelson Maculan, Thiago F Noronha, and Celso C Ribeiro. A branch-and-cut algorithm for partition coloring. Networks: An International Journal, 55(3):194-204, 2010. Google Scholar
  17. Fabio Furini, Enrico Malaguti, and Alberto Santini. An exact algorithm for the partition coloring problem. Computers & Operations Research, 92:170-181, 2018. URL: https://doi.org/10.1016/j.cor.2017.12.019.
  18. Bruce Golden, Saahitya Raghavan, and Daliborka Stanojevic. The prize-collecting generalized minimum spanning tree problem. Journal of Heuristics, 14:69-93, February 2008. URL: https://doi.org/10.1007/s10732-007-9027-1.
  19. Mohamed Haouari and Jouhaina Siala. Upper and lower bounding strategies for the generalized minimum spanning tree problem. European Journal of Operational Research, 171:632-647, February 2006. URL: https://doi.org/10.1016/j.ejor.2004.07.072.
  20. Alain Hertz, Odile Marcotte, and David Schindl. On the maximum orders of an induced forest, an induced tree, and a stable set. Yugoslav Journal of Operations Research, 24:199-215, January 2014. URL: https://doi.org/10.2298/YJOR130402037H.
  21. Edmund Ihler, Gabriele Reich, and Peter Widmayer. Class Steiner trees and VLSI-design. Discrete Applied Mathematics, 90(1):173-194, 1999. URL: https://doi.org/10.1016/S0166-218X(98)00090-0.
  22. Michal Karonski and Zbigniew Palka. On the size of a maximal induced tree in a random graph. Math. Slovaca, 30:151-155, 1980. Google Scholar
  23. Gilbert Laporte, Ardavan Asef-Vaziri, and Chelliah Sriskandarajah. Some applications of the generalized travelling salesman problem. The Journal of the Operational Research Society, 47(12):1461-1467, 1996. Google Scholar
  24. Gilbert Laporte and François V. Louveaux. The integer L-shaped method for stochastic integer programs with complete recourse. Operations Research Letters, 13(3):133-142, 1993. URL: https://doi.org/10.1016/0167-6377(93)90002-X.
  25. Gilbert Laporte and Yves Nobert. Generalized travelling salesman problem through n sets of nodes: An integer programming approach. INFOR: Information Systems and Operational Research, 21(1):61-75, 1983. URL: https://doi.org/10.1080/03155986.1983.11731885.
  26. Guangzhi Li and Rahul Simha. The partition coloring problem and its application to wavelength routing and assignment. In In Proceedings of the First Workshop on Optical Networks, 2000. Google Scholar
  27. Rafael A. Melo and Celso C. Ribeiro. Maximum weighted induced forests and trees: New formulations and a computational comparative review. International Transactions in Operational Research, 2021. URL: https://doi.org/10.1111/itor.13066.
  28. Young Soo Myung, Chang Ho Lee, and Dong Wan Tcha. On the generalized minimum spanning tree problem. Networks, 26(4):231-241, 1995. URL: https://doi.org/10.1002/net.3230260407.
  29. Charles E. Noon and James C. Bean. A lagrangian based approach for the asymmetric generalized traveling salesman problem. Operations Research, 39(4):623-632, 1991. Google Scholar
  30. Temel Oncan, Jean-François Cordeau, and Gilbert Laporte. A tabu search heuristic for the generalized minimum spanning tree problem. European Journal of Operational Research, 191:306-319, December 2008. URL: https://doi.org/10.1016/j.ejor.2007.08.021.
  31. Petrica C. Pop. On the prize-collecting generalized minimum spanning tree problem. Annals of Operations Research, 150:193-204, March 2007. URL: https://doi.org/10.1007/s10479-006-0153-1.
  32. Petrica C. Pop. The generalized minimum spanning tree problem: An overview of formulations, solution procedures and latest advances. European Journal of Operational Research, 283(1):1-15, 2020. URL: https://doi.org/10.1016/j.ejor.2019.05.017.
  33. Petrica C. Pop, Bin Hu, and Günther Raidl. A memetic algorithm with two distinct solution representations for the partition graph coloring problem. In Revised Selected Papers of the 14th International Conference on Computer Aided Systems Theory - EUROCAST 2013 - Volume 8111, volume 8111, pages 219-226. Springer-Verlag, February 2013. URL: https://doi.org/10.1007/978-3-642-53856-8_28.
  34. Petrica C. Pop, Walter Kern, and Georg J. Still. The Generalized Minimum Spanning Tree Problem. University of Twente, Department of Applied Mathematics, 2000. Google Scholar
  35. Petrica C. Pop, Oliviu Matei, Cosmin Sabo, and Adrian Petrovan. A two-level solution approach for solving the generalized minimum spanning tree problem. European Journal of Operational Research, August 2017. URL: https://doi.org/10.1016/j.ejor.2017.08.015.
  36. Dieter Rautenbach. Dominating and large induced trees in regular graphs. Discrete Mathematics, 307(24):3177-3186, 2007. URL: https://doi.org/10.1016/j.disc.2007.03.043.
  37. Gabriele Reich and Peter Widmayer. Beyond Steiner’s problem: A VLSI oriented generalization. In Graph-Theoretic Concepts in Computer Science, volume 411 of Lecture Notes in Computer Science, pages 196-210. Springer, Berlin, Heidelberg, 1990. URL: https://doi.org/10.1007/3-540-52292-1_14.
  38. Oylum Şeker, Tınaz Ekim, and Z. Caner Taşkın. A decomposition approach to solve the selective graph coloring problem in some perfect graph families. Networks, 73(2):145-169, 2019. URL: https://doi.org/10.1002/net.21850.
  39. Oylum Şeker, Tınaz Ekim, and Z. Caner Taşkın. An exact cutting plane algorithm to solve the selective graph coloring problem in perfect graphs. European Journal of Operational Research, 291(1):67-83, 2021. URL: https://doi.org/10.1016/j.ejor.2020.09.017.
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