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Mixed Integer Programming Formulations for Steiner Tree and Quality of Service Multicast Tree Problems

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Abstract

This article presents flow-based mixed integer programming formulations for the Steiner Tree Problem and its variant applied to model and solve the Quality of Service Multicast Tree problem. This is a relevant problem related to nowadays telecommunication networks, particularly Content Delivery Networks, to distribute multimedia over cloud-based Internet systems. To the best of our knowledge, no previous mixed integer programming formulation was proposed for the Quality of Service Multicast Tree problem variant. Experimental evaluation is performed over a set of realistic problem instances from SteinLib, a reference test-set repository, modified accordingly to prove that standard exact solvers are capable of finding solutions to real-world size instances. Exact methods are applied for benchmarking the proposed formulations, finding optimal solutions and low feasible-to-optimal gaps in reasonable execution times.

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REFERENCES

  1. Ahmed, R., Watkins, J., Wolff, A., Angelini, P., Darabi, F., Efrat, A., Glickenstein, D., Gronemann, M., Heinsohn, N., Kobourov, S., and Spence, R., Multi-level Steiner trees, J. Experim. Algorithmics, 2019, vol. 24, no. 1.

  2. Aneja, Y., An integer linear programming approach to the Steiner problem in graphs, Networks, 1980, vol. 10, no. 2, pp. 167–178.

    Article  MathSciNet  Google Scholar 

  3. Angelopoulos, S., A near-tight bound for the online steiner tree problem in graphs of bounded asymmetry, in Proc. European Symp, on Algorithms, Berlin, Heidelberg: Springer, 2008, pp. 76–87.

  4. The Oxford Handbook of Information and Communication Technologies, Avgerou, C., Mansell, R., Quah, D., and Silverstone, R., Eds., Oxford Univ. Press, 2009.

    Google Scholar 

  5. Beasley, J., An SST-based algorithm for the Steiner problem in graphs, Networks, 1989, vol. 19, no. 1, pp. 1–16.

    Article  MathSciNet  Google Scholar 

  6. Borradaile, G., Klein, P., and Mathieu, C., An O(nlogn) approximation scheme for Steiner tree in planar graphs, ACM Trans. Algorithms, 2009, vol. 5, no. 3, pp. 1–31.

    Article  MathSciNet  Google Scholar 

  7. Chimani, M., Kandyba, M., Ljubic, I., and Mutzel, P., Strong formulations for 2-node-connected Steiner network problems, Proc. Combinatorial Optimization and Applications, St. John’s, 2008, pp. 190–200.

    Book  Google Scholar 

  8. Duin, C. and Voß, S., Steiner tree heuristics – a survey, in Operations Research Proc., 1993, pp. 485–496.

  9. Ford, L. and Fulkerson, D., Flows in Networks, Princeton: Princeton Univ. Press, 2010.

    MATH  Google Scholar 

  10. Freeman, R., Telecommunication System Engineering, John Wiley & Sons Inc., 2004.

    Book  Google Scholar 

  11. Guanyu Gao, Weiwen Zhang, Yong Wen, Zhi Wang, and Wenwu Zhu, Towards cost-efficient video transcoding in media cloud: insights learned from user viewing patterns, IEEE Trans. Multimedia, 2015, vol. 17, no. 8, pp. 1286–1296.

    Article  Google Scholar 

  12. Gelenbe, E., Ghanwani, A., and Srinivasan, V., Improved neural heuristics for multicast routing, IEEE J. Select. Areas Commun., 1997, vol. 15, no. 2, pp. 147–155.

    Article  Google Scholar 

  13. Goemans M. and Young-Soo Myung, A catalog of Steiner tree formulations, Networks, 1993, vol. 23, no. 1, pp. 19–28.

    Article  MathSciNet  Google Scholar 

  14. Hauptmann, M. and Karpinski, M., A compendium on Steiner tree problems, Tech. Rep., Department of Computer Science and Hausdorff Center for Mathematics Univ. of Bonn, 2014.

    Google Scholar 

  15. Menglan Hu, Jun Luo, Yang Wang, and Bharadwa j Veeravalli, Practical resource provisioning and caching with dynamic resilience for cloud-based content distribution networks, IEEE Trans. Parallel Distribut. Syst., 2014, vol. 25, no. 8, pp. 2169–2179.

  16. ILOG CPLEX. Terminating MIP Optimization. http://www.eio.upc.es/lceio/manuals/cplex-11/html/usrcplex/solveMIP10.html. Jan. 2020.

  17. Iturriaga, S., Nesmachnow, S., Goñi, G., Dorronsoro, B., and Tchernykh, A., Evolutionary algorithms for optimizing cost and QoS on cloud-based content distribution networks, Program. Comput. Software, 2019, vol. 45, pp. 544–556.

    Article  Google Scholar 

  18. Narendra Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 1984, vol. 4, no. 4, pp. 373–395.

    Article  MathSciNet  Google Scholar 

  19. Karp, R., Reducibility among combinatorial problems, in Complexity of Computer Computations, Miller, R. and Thatcher, J., Eds., New York: Plenum Press, 1972, pp. 85–103.

    Google Scholar 

  20. Karpinski, M., Mandoiu, I., Olshevsky, A., and Zelikovsky, A., Improved approximation algorithms for the quality of service multicast tree problem, Algorithmica, 2005, vol. 42, no. 2, pp. 109–120.

    Article  MathSciNet  Google Scholar 

  21. Khallef, W., Durand, S., and Molnár, M., ILP formulation of the exact solution of multi-constrained minimum cost multicast, Comput. Networks, 2018, vol. 135, pp. 160–170.

    Article  Google Scholar 

  22. Moonseong Kim, Hyunseung Choo, Matt Mutka, Hyung-Jin Lim, and Kwangjin Park, On QoS multicast routing algorithms using k-minimum Steiner trees, Inf. Sci., 2013, vol. 238, pp. 190–204.

    Article  MathSciNet  Google Scholar 

  23. Koch, T. and Martin, A., Solving Steiner tree problems in graphs to optimality, Networks, 1998, vol. 32, no. 3, pp. 207–232.

    Article  MathSciNet  Google Scholar 

  24. Koch, T., Martin, A., and Voß, S., SteinLib: an updated library on Steiner tree problems in graphs, in Combinatorial Optimization, Springer US, 2001, pp. 285–325.

  25. Lawler, E., Combinatorial Optimization: Networks and Matroids, Dover Publ., 2011.

    MATH  Google Scholar 

  26. Leggieri, V., Haouari, M., and Triki, C., The Steiner tree problem with delays: a compact formulation and reduction procedures, Discrete Appl. Math., 2014, vol. 164, pp. 178–190.

    Article  MathSciNet  Google Scholar 

  27. Lei Liu, Hua Wang, and Guohong Kong, An improved EDA for solving Steiner tree problem, Concurrency Comput.: Pract. Exper., 2015, vol. 27, no. 13, pp. 3483–3496.

    Article  Google Scholar 

  28. Martins, S., Resende, M., Ribeiro, C., and Pardalos, P., J. Global Optim., 2000, vol. 17, no. 1/4, pp. 267–283.

    Article  MathSciNet  Google Scholar 

  29. Minoux, M., Efficient greedy heuristics for Steiner tree problems using reolptimization and super modularity, Inf. Syst. Oper. Res., 1990, vol. 28, no. 3, pp. 221–233.

    MATH  Google Scholar 

  30. Monma, C., Munson, B., and Pulleyblank, W., Minimum-weight two-connected spanning networks, Math. Program., 1990, vol. 46, no. 1–3, pp. 153–171.

    Article  MathSciNet  Google Scholar 

  31. Nesmachnow, S., An overview of metaheuristics: accurate and efficient methods for optimisation, Int. J. Metaheuristics, 2014, vol. 3, no. 4, p. 320.

    Article  Google Scholar 

  32. Polzin, T. and Daneshmand, S., Improved algorithms for the Steiner problem in networks, Discrete Appl. Math., 2001, vol. 112, no. 1–3, pp. 263–300.

    Article  MathSciNet  Google Scholar 

  33. Ribeiro, C., Uchoa, E., and Werneck, R., J. Comput., 2002, vol. 14, no. 3, pp. 228–246.

    Google Scholar 

  34. Siebert, M., Ahmed, S., and Nemhauser, G., A linear programming based approach to the Steiner tree problem with a fixed number of terminals, Networks, 2020, vol. 75, no. 2.

  35. Stanojevic, M. and Vujosevic, M., An exact algorithm for Steiner tree problem on graphs, Int. J. Comput., Commun. Control, 2006, vol. 1, no. 1, pp. 41–46.

    Article  Google Scholar 

  36. Xinhui Wang, Exact algorithms for the Steiner tree problem, PhD Thesis, Department of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science of the University of Twente, 2008.

  37. Wong, R., A dual ascent approach for Steiner tree problems on a directed graph, Math. Program., 1984, vol. 28, no. 3, pp. 271–287.

    Article  MathSciNet  Google Scholar 

  38. Wenhua Xiao, Weidong Bao, Xiaomin Zhu, Chen Wang, Lidong Chen, and Yang, L.T., Dynamic request redirection and resource provisioning for cloud-based video services under heterogeneous environment, IEEE Trans. Parallel Distribut. Syst., 2016, vol. 27, no. 7, pp. 1954–1967.

    Article  Google Scholar 

  39. Weijun Yang and Yuanfeng Chen, A novel QoS provisioning algorithm for optimal ulticast routing in WMNs, Future Internet, 2016, vol. 8, no. 3.

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  1. Universidad de la República, 11200, Montevideo, Departamento de Montevideo, Uruguay

    C. Risso, F. Robledo & S. Nesmachnow

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  1. C. Risso
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  2. F. Robledo
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Risso, C., Robledo, F. & Nesmachnow, S. Mixed Integer Programming Formulations for Steiner Tree and Quality of Service Multicast Tree Problems. Program Comput Soft 46, 661–678 (2020). https://doi.org/10.1134/S0361768820080174

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