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The symmetric quadratic traveling salesman problem

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Abstract

In the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession. This structure arises, e.g., if the succession of two edges represents energetic conformations, a change of direction or a possible change of transportation means. In the symmetric case, costs do not depend on the direction of traversal. We study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem. Our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities. We establish relations to facets of the Boolean quadric polytope, exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges, and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem. Applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities, but finding a maximally violated inequality among these is NP-complete. For the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet. Preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications.

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References

  1. Achterberg T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aggarwal A., Coppersmith D., Khanna S., Motwani R., Schieber B.: The angular-metric traveling salesman problem. SIAM J. Comput. 29, 697–711 (1999)

    Article  MathSciNet  Google Scholar 

  3. Amaldi E., Galbiati G., Maffioli F.: On minimum reload cost paths, tours, and flows. Networks 57, 254–260 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Applegate D.L., Bixby R.E., Chvatal V., Cook W.J.: The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics). Princeton University Press, Princeton (2007)

    Google Scholar 

  5. Billionnet A., Elloumi S.: Using a mixed integer quadratic programming solver for the unconstrained quadratic 0–1 problem. Math. Program. 109, 55–68 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Billionnet A., Elloumi S., Plateau M.C.: Improving the performance of standard solvers for quadratic 0–1 programs by a tight convex reformulation: the QCR method. Discrete Appl. Math. 157, 1185–1197 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burkard, R., Çela, E., Klinz, B.: On the biquadratic assignment problem. In: Pardalos P., Wolkowicz, H. (eds.) Quadratic Assignment and Related Problems. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16. pp. 117–146 (1994)

  8. Chvátal V.: Edmonds polytopes and weakly hamiltonian graphs. Math. Program. 5, 29–40 (1973)

    Article  MATH  Google Scholar 

  9. Dantzig G., Fulkerson R., Johnson S.: Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)

    Article  MathSciNet  Google Scholar 

  10. De Klerk E., Pasechnik D.V., Sotirov R.: On semidefinite programming relaxations of the traveling salesman problem. SIAM J. Optim. 19(4), 1559–1573 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Faigle U., Frahling G.: A combinatorial algorithm for weighted stable sets in bipartite graphs. Discrete Appl. Math. 154(9), 1380–1391 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fischer, F., Jäger, G., Lau, A., Molitor, P.: Complexity and Algorithms for the Traveling Salesman Problem and the Assignment Problem of Second Order. Preprint 2009-16, Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany (2009)

  13. Galbiati G., Gualandi S., Maffioli F.: On minimum reload cost cycle cover. Electron. Notes Discrete Math. 36, 81–88 (2010)

    Article  Google Scholar 

  14. Gourvès L., Lyra A., Martinhon C., Monnot J.: The minimum reload s-t path, trail and walk problems. Discrete Appl. Math. 158, 1404–1417 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grötschel M., Padberg M.W.: Lineare Charakterisierungen von Travelling Salesman Problemen. Zeitschrift für Oper. Res. Ser. A 21(1), 33–64 (1977)

    MATH  Google Scholar 

  16. Grötschel M., Padberg M.W.: On the symmetric travelling salesman problem I: inequalities. Math. Program. 16, 265–280 (1979)

    Article  MATH  Google Scholar 

  17. Grötschel M., Padberg M.W.: On the symmetric travelling salesman problem II: lifting theorems and facets. Math. Program. 16, 281–302 (1979)

    Article  MATH  Google Scholar 

  18. Gutin G., Punnen A., Barvinok A., Gimadi E.K., Serdyukov A.I.: The Traveling Salesman Problem and Its Variations (Combinatorial Optimization). Springer, Berlin (2002)

    Google Scholar 

  19. Hong, S.: A Linear Programming Approach for the Traveling Salesman Problem. Ph.D. thesis, John Hopkins University, Baltimore, MD, USA (1972)

  20. IBM ILOG CPLEX 12.2: Using the CPLEX callable library. Information available at http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/

  21. Jäger G., Molitor P.: Algorithms and experimental study for the traveling salesman problem of second order. Lect. Notes Comput. Sci. 5165, 211–224 (2008)

    Article  Google Scholar 

  22. Lawler, E.L., Lenstra, J.K., Kan, A.H.G.R., Shmoys, D.B. (eds): The Traveling Salesman Problem. A Guided Tour of Combinatorial Optimization. Wiley, Chichester (1985)

    MATH  Google Scholar 

  23. Lovász L., Schrijver A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mathematica 7. Information available at http://www.wolfram.com/mathematica/

  25. Padberg M.: The Boolean quadric polytope: some characteristics, facets and relatives. Math. Program. 45, 139–172 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. SCIP 2.1.1. Information available at http://scip.zib.de/

  27. Shor, N.Z.: Quadratic optimization problems. Sov. J. Comput. Syst. Sci. 25, 1–11 (1987). Originally published in Tekhnicheskaya Kibernetika, No. 1, 1987, pp. 128–139

    Google Scholar 

  28. Wirth H.C., Steffan J.: On minimum diameter spanning trees under reload costs. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds) Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 1665, pp. 78–89. Springer, Berlin (1999)

    Google Scholar 

  29. Zhao X., Huang H., Speed T.P.: Finding short DNA motifs using permuted markov models. J. Comput. Biol. 12(6), 894–906 (2005)

    Article  Google Scholar 

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Correspondence to Anja Fischer.

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Fischer, A., Helmberg, C. The symmetric quadratic traveling salesman problem. Math. Program. 142, 205–254 (2013). https://doi.org/10.1007/s10107-012-0568-1

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