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Implications, Conflicts, and Reductions for Steiner Trees

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Integer Programming and Combinatorial Optimization (IPCO 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12707))

Abstract

The Steiner tree problem in graphs (SPG) is one of the most studied problems in combinatorial optimization. In the last decade, there have been significant advances concerning approximation and complexity of the SPG. However, the state of the art in (practical) exact solution of the SPG has remained largely unchallenged for almost 20 years.

The following article seeks to once again advance exact SPG solution. The article is based on a combination of three concepts: Implications, conflicts, and reductions. As a result, various new SPG techniques are conceived. Notably, several of the resulting techniques are provably stronger than well-known methods from the literature, used in exact SPG algorithms. Finally, by integrating the new methods into a branch-and-cut algorithm we obtain an exact SPG solver that outperforms the current state of the art on a large collection of benchmark sets. Furthermore, we can solve several instances for the first time to optimality.

Supported by Research Campus Modal, and DFG Cluster of Excellence MATH+.

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References

  1. Achterberg, T.: Constraint Integer Programming. Ph.D. thesis, Technische Universität Berlin (2007)

    Google Scholar 

  2. Bonnet, É., Sikora, F.: The PACE 2018 parameterized algorithms and computational experiments challenge: the third iteration. In: Paul, C., Pilipczuk, M. (eds.) 13th International Symposium on Parameterized and Exact Computation (IPEC 2018), Leibniz International Proceedings in Informatics (LIPIcs), vol. 115, pp. 26:1–26:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2019). https://doi.org/10.4230/LIPIcs.IPEC.2018.26

  3. Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: Steiner tree approximation via iterative randomized rounding. J. ACM 60(1), 6 (2013). https://doi.org/10.1145/2432622.2432628

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheng, X., Du, D.Z.: Steiner trees in industry. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, vol. 11, pp. 193–216. Springer, Boston https://doi.org/10.1007/0-387-23830-1_4

  5. Duin, C.: Steiner Problems in Graphs. Ph.D. thesis, University of Amsterdam (1993)

    Google Scholar 

  6. Duin, C.W., Volgenant, A.: Reduction tests for the Steiner problem in graphs. Networks 19(5), 549–567 (1989). https://doi.org/10.1002/net.3230190506

    Article  MathSciNet  MATH  Google Scholar 

  7. Duin, C., Volgenant, A.: An edge elimination test for the Steiner problem in graphs. Oper. Re. Lett. 8(2), 79–83 (1989). https://doi.org/10.1016/0167-6377(89)90005-9

    Article  MathSciNet  MATH  Google Scholar 

  8. Fischetti, M., et al.: Thinning out Steiner trees: a node-based model for uniform edge costs. Math. Program. Comput. 9(2), 203–229 (2016). https://doi.org/10.1007/s12532-016-0111-0

    Article  MathSciNet  MATH  Google Scholar 

  9. Gamrath, G., Koch, T., Maher, S.J., Rehfeldt, D., Shinano, Y.: SCIP-Jack—a solver for STP and variants with parallelization extensions. Math. Program. Comput. 9(2), 231–296 (2016). https://doi.org/10.1007/s12532-016-0114-x

    Article  MathSciNet  MATH  Google Scholar 

  10. Goemans, M.X., Olver, N., Rothvoß, T., Zenklusen, R.: Matroids and integrality gaps for Hypergraphic Steiner tree relaxations. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, STOC 2012, pp. 1161–1176. Association for Computing Machinery, New York, NY, USA (2012). https://doi.org/10.1145/2213977.2214081

  11. Hwang, F., Richards, D., Winter, P.: The Steiner Tree Problem. Elsevier Science, Annals of Discrete Mathematics (1992)

    Google Scholar 

  12. IBM: Cplex (2020). https://www.ibm.com/analytics/cplex-optimizer

  13. Iwata, Y., Shigemura, T.: Separator-based pruned dynamic programming for steiner tree. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 33, pp. 1520–1527 (2019). https://doi.org/10.1609/aaai.v33i01.33011520

  14. Juhl, D., Warme, D.M., Winter, P., Zachariasen, M.: The GeoSteiner software package for computing Steiner trees in the plane: an updated computational study. Math. Program. Comput. 10(4), 487–532 (2018). https://doi.org/10.1007/s12532-018-0135-8

    Article  MathSciNet  MATH  Google Scholar 

  15. Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972)

    Google Scholar 

  16. Kisfaludi-Bak, S., Nederlof, J., Leeuwen, E.J.V.: Nearly ETH-tight algorithms for planar Steiner tree with terminals on few faces. ACM Trans. Algorithms (TALG) 16(3), 1–30 (2020). https://doi.org/10.1145/3371389

  17. Koch, T., Martin, A.: Solving Steiner tree problems in graphs to optimality. Networks 32, 207–232 (1998). https://doi.org/10.1002/(SICI)1097-0037(199810)32:3%3C207::AID-NET5%3E3.0.CO;2-O

  18. Koch, T., Martin, A., Voß, S.: SteinLib: An updated library on Steiner tree problems in graphs. In: Du, D.Z., Cheng, X. (eds.) Steiner Trees in Industries, pp. 285–325. Kluwer (2001)

    Google Scholar 

  19. Leitner, M., Ljubic, I., Luipersbeck, M., Prossegger, M., Resch, M.: New Real-world Instances for the Steiner Tree Problem in Graphs. Technical Report, ISOR, Uni Wien (2014)

    Google Scholar 

  20. Pajor, T., Uchoa, E., Werneck, R.F.: A robust and scalable algorithm for the Steiner problem in graphs. Math. Program. Comput. 10(1), 69–118 (2017). https://doi.org/10.1007/s12532-017-0123-4

    Article  MathSciNet  MATH  Google Scholar 

  21. Polzin, T.: Algorithms for the Steiner problem in networks. Ph.D. thesis, Saarland University (2003)

    Google Scholar 

  22. Polzin, T., Daneshmand, S.V.: A comparison of Steiner tree relaxations. Discrete Appl. Math. 112(1–3), 241–261 (2001)

    Article  MathSciNet  Google Scholar 

  23. Polzin, T., Daneshmand, S.V.: Improved Algorithms for the Steiner Problem in Networks. Discrete Appl. Math. 112(1–3), 263–300 (2001). https://doi.org/10.1016/S0166-218X(00)00319-X

    Article  MathSciNet  MATH  Google Scholar 

  24. Polzin, T., Daneshmand, S.V.: Extending reduction techniques for the Steiner tree problem. In: Möhring, R., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 795–807. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45749-6_69

    Chapter  Google Scholar 

  25. Polzin, T., Daneshmand, S.V.: On Steiner trees and minimum spanning trees in hypergraphs. Oper. Res. Lett. 31(1), 12–20 (2003). https://doi.org/10.1016/S0167-6377(02)00185-2

    Article  MathSciNet  MATH  Google Scholar 

  26. Polzin, T., Daneshmand, S.V.: Practical partitioning-based methods for the Steiner problem. In: Àlvarez, C., Serna, M. (eds.) WEA 2006. LNCS, vol. 4007, pp. 241–252. Springer, Heidelberg (2006). https://doi.org/10.1007/11764298_22

    Chapter  Google Scholar 

  27. Polzin, T., Vahdati-Daneshmand, S.: The Steiner Tree Challenge: An updated Study (2014), unpublished manuscript at http://dimacs11.cs.princeton.edu/downloads.html

  28. Rehfeldt, D., Koch, T.: Implications, conflicts, and reductions for Steiner trees. Technical Report 20–28, ZIB, Takustr. 7, 14195 Berlin (2020)

    Google Scholar 

  29. Rehfeldt, D., Shinano, Y., Koch, T.: SCIP-jack: an exact high performance solver for Steiner tree problems in graphs and related problems. In: Bock, H.G., Jäger, W., Kostina, E., Phu, H.X. (eds.) Modeling, Simulation and Optimization of Complex Processes HPSC 2018. LNCS, pp. 201–223. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-55240-4_10

    Chapter  Google Scholar 

  30. Rosseti, I., de Aragão, M., Ribeiro, C., Uchoa, E., Werneck, R.: New benchmark instances for the Steiner problem in graphs. In: Extended Abstracts of the 4th Metaheuristics International Conference (MIC 2001), pp. 557–561. Porto (2001)

    Google Scholar 

  31. Takahashi, H., Matsuyama, A.: An approximate solution for the Steiner problem in graphs. Math. Japonicae 24, 573–577 (1980)

    MathSciNet  MATH  Google Scholar 

  32. Uchoa, E., Poggi de Aragão, M., Ribeiro, C.C.: Preprocessing Steiner problems from VLSI layout. Networks 40(1), 38–50 (2002). https://doi.org/10.1002/net.10035

  33. Vahdati Daneshmand, S.: Algorithmic approaches to the Steiner problem in networks. Ph.D. thesis, Universität Mannheim (2004)

    Google Scholar 

  34. Vygen, J.: Faster algorithm for optimum Steiner trees. Inf. Process. Lett. 111(21), 1075–1079 (2011). https://doi.org/10.1016/j.ipl.2011.08.005

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Chair of Software and Algorithms for Discrete Optimization, TU Berlin, Straße des 17. Juni 135, 10623, Berlin, Germany

    Daniel Rehfeldt & Thorsten Koch

  2. Applied Algorithmic Intelligence Methods Department, Zuse Institute Berlin, Takustraße 7, 14195, Berlin, Germany

    Daniel Rehfeldt & Thorsten Koch

Authors
  1. Daniel Rehfeldt
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  2. Thorsten Koch
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Correspondence to Daniel Rehfeldt .

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Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Mohit Singh

  2. Cornell University, Ithaca, NY, USA

    David P. Williamson

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Rehfeldt, D., Koch, T. (2021). Implications, Conflicts, and Reductions for Steiner Trees. In: Singh, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2021. Lecture Notes in Computer Science(), vol 12707. Springer, Cham. https://doi.org/10.1007/978-3-030-73879-2_33

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  • DOI: https://doi.org/10.1007/978-3-030-73879-2_33

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  • Print ISBN: 978-3-030-73878-5

  • Online ISBN: 978-3-030-73879-2

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