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Semidefinite relaxations of ordering problems

Abstract

Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element \(u\) is before \(v\) in the ordering. In the second case, the profit depends on whether \(u\) is before \(v\) and \(r\) is before \(s\). The linear ordering problem is well studied, with exact solution methods based on polyhedral relaxations. The quadratic ordering problem does not seem to have attracted similar attention. We present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem, extending and improving existing approaches. We show the efficiency of our relaxations by providing computational experience on a variety of problem classes.

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Notes

  1. 1.

    For exact numbers of the speed differences see http://www.cpubenchmark.net/.

  2. 2.

    These instances can be downloaded from http://flplib.uwaterloo.ca/.

References

  1. 1.

    Achatz, H., Kleinschmidt, P., Lambsdorff, J.: Der corruption perceptions index und das linear ordering problem. ORNews 26, 10–12 (2006)

    Google Scholar 

  2. 2.

    Amaral, A.R.S.: A new lower bound for the single row facility layout problem. Discrete Appl. Math. 157(1), 183–190 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Amaral, A.R.S., Letchford, A.N.: A polyhedral approach to the single row facility layout problem (in preparation). Preprint available from http://www.optimization-online.org/DB_FILE/2008/03/1931.pdf (2011)

  4. 4.

    Anjos, M.F., Kennings, A., Vannelli, A.: A semidefinite optimization approach for the single-row layout problem with unequal dimensions. Discrete Optim. 2(2), 113–122 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Anjos, M.F., Vannelli, A.: Computing globally optimal solutions for single-row layout problems using semidefinite programming and cutting planes. INFORMS J. Comput. 20(4), 611–617 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Anjos, M.F., Yen, G.: Provably near-optimal solutions for very large single-row facility layout problems. Optim. Methods Softw. 24(4), 805–817 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Boenchendorf, K.: Reihenfolgenprobleme/Mean-flow-time sequencing. Mathematical Systems in Economics 74. Verlagsgruppe Athenaum, Hain, Scriptor (1982)

  8. 8.

    Buchheim, C., Wiegele, A., Zheng, L.: Exact algorithms for the quadratic linear ordering problem. INFORMS J. Comput. 22(1), 168–177 (2009)

    Google Scholar 

  9. 9.

    Caprara, A., Jung M., Oswald, M., Reinelt, G., Traversi, E.: A betweenness approach for solving the linear arrangement problem. Technical report, University of Heidelberg (2011, in preparation)

  10. 10.

    Caprara, A., Letchford, A.N., Salazar-Gonzalez, J.-J.: Decorous lower bounds for minimum linear arrangement. INFORMS J. Comput. 23(1), 26–40 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Chenery, H., Watanabe, T.: International comparisons of the structure of production. Econometrica 26, 487–521 (1958)

    Article  Google Scholar 

  12. 12.

    Chimani, M., Hungerländer, P., Jünger, M., Mutzel, P.: An SDP approach to multi-level crossing minimization. In: Proceedings of Algorithm Engineering & Experiments [ALENEX’2011] (2011)

  13. 13.

    Christof, T., Oswald, M., Reinelt, G.: Consecutive ones and a betweenness problem in computational biology. In: Proceedings of the 6th Conference on Integer Programming and Combinatorial Optimization (IPCO 1998). Lecture Notes in Computer Science, vol. 1412, pp. 213–228. Springer, Berlin (1998)

  14. 14.

    Duff, I.S., Grimes, R.G., Lewis, J.G.: Users’ Guide for the Harwell-Boeing Sparse Matrix Collection. Technical report, CERFACS, Toulouse, France (1992)

  15. 15.

    Fischer, I., Gruber, G., Rendl, F., Sotirov, R.: Computational experience with a bundle method for semidefinite cutten plane relaxations of max-cut and equipartition. Math. Program. 105, 451–469 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified np-complete problems. In: STOC ’74: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, pp. 47–63, New York (1974)

  17. 17.

    Glover, F., Klastorin, T., Klingman, D.: Optimal weighted ancestry relationships. Manag. Sci. 20, 1190–1193 (1974)

    MATH  Article  Google Scholar 

  18. 18.

    Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Grötschel, M., Jünger, M., Reinelt, G.: A cutting plane algorithm for the linear ordering problem. Oper. Res. 32(6), 1195–1220 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Harper, L.H.: Optimal assignments of numbers to vertices. SIAM J. Appl. Math. 12(1), 131–135 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Healy, P., Kuusik, A.: The vertex-exchange graph: a new concept for multilevel crossing minimisation. In: Proceedings of the Symposium on Graph Drawing [GD’99], pp. 205–216. Springer, Berlin (1999)

  22. 22.

    Helmberg, C.: Fixing variables in semidefinite relaxations. SIAM J. Matrix Anal. Appl. 21(3), 952–969 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Helmberg, C., Mohar, B., Poljak, S., Rendl, F.: A spectral approach to bandwidth and separator problems in graphs. Linear Multilinear Algebra 39, 73–90 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Helmberg, C., Rendl, F., Vanderbei, R., Wolkowicz, H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6, 342–361 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Heragu, S.S., Kusiak, A.: Machine layout problem in flexible manufacturing systems. Oper. Res. 36(2), 258–268 (1988)

    Article  Google Scholar 

  26. 26.

    Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms (vol. 1 and 2). Springer, Berlin (1993)

  27. 27.

    Hungerländer, P.: Semidefinite Approaches to Ordering Problems. PhD thesis, Alpen-Adria Universität Klagenfurt (2012)

  28. 28.

    Hungerländer, P., Rendl, F.: A computational study and survey of methods for the single-row facility layout problem. Comput. Optim. Appl. (2012, accepted)

  29. 29.

    Jünger, M., Mutzel, P.: 2-Layer straightline crossing minimization: performance of exact and heuristic algorithms. J. Graph Algorithms Appl. 1, 1–25 (1997)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Juvan, M., Mohar, B.: Optimal linear labelings and eigenvalues of graphs. Discrete Appl. Math. 36(2), 153–168 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Kaas, R.: A branch and bound algorithm for the acyclic subgraph problem. Eur. J. Oper. Res. 8(4), 355–362 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Knuth, D.E.: The Stanford GraphBase: A Platform for Combinatorial Computing. ACM, New York (1993)

    Google Scholar 

  33. 33.

    Leontief, W.: Quantitative input-output relations in the economic system of the united states. Rev. Econ. Stat. 18(3), 105–125 (1936)

    Google Scholar 

  34. 34.

    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Mart, R., Reinelt, G., Duarte, A.: A benchmark library and a comparison of heuristic methods for the linear ordering problem. Comput. Optim. Appl. 51(3), 1–21 (2011)

    Google Scholar 

  36. 36.

    Martí, R., Laguna, M.: Heuristics and meta-heuristics for 2-layer straight line crossing minimization. Discrete Appl. Math. 127(3), 665–678 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Mitchell, J.E., Borchers, B.: Solving linear ordering problems with a combined interior pointsimplex cutting plane algorithm. In: Frenk, T.T.H., Roos, K., Zhang, S. (eds.) High Performance Optimization, pp. 349–366. Kluwer, Dordrecht (2000)

  38. 38.

    Newman, A.: Cuts and orderings: on semidefinite relaxations for the linear ordering problem. In: Jansen, K., Khanna, S., Rolim, J., Ron, D. (eds.) Lecture Notes in Computer Science, vol. 3122, pp. 195–206. Springer, Berlin (2004)

  39. 39.

    Schwarz, R.: A Branch-and-Cut Algorithm with Betweenness Variables for the Linear Arrangement Problems. Diploma thesis, Heidelberg (2010)

  40. 40.

    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3(3), 411–430 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Simmons, D.M.: One-dimensional space allocation: an ordering algorithm. Oper. Res. 17, 812–826 (1969)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Simone, C.D.: The cut polytope and the Boolean quadric polytope. Discrete Math. 79(1), 71–75 (1990)

    MATH  Article  Google Scholar 

  43. 43.

    Sturm, J.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones (updated for version 1.05). Available under http://sedumi.ie.lehigh.edu (2001)

  44. 44.

    Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst. Man Cybern. 11(2), 109–125 (1981)

    MathSciNet  Article  Google Scholar 

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Acknowledgments

We thank Gerhard Reinelt and Marcus Oswald from Universität Heidelberg for many fruitful discussions and practical hints regarding the polyhedral approach to LOP. We also acknowledge constructive comments by two anonymous referees that led to the current version of the paper.

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Correspondence to F. Rendl.

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This paper is dedicated to Claude Lemarechal on the occasion of his 65th birthday.

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Hungerländer, P., Rendl, F. Semidefinite relaxations of ordering problems. Math. Program. 140, 77–97 (2013). https://doi.org/10.1007/s10107-012-0627-7

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Mathematics Subject Classification

  • 90C22