Skip to main content
Log in

Continuous relaxations for the traveling salesman problem

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript


In this work, we aim to explore connections between dynamical systems techniques and combinatorial optimization problems. In particular, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the relaxed discrete optimization problem into appropriate manifolds. We explore multiple embedding techniques—namely the construction of new dynamical systems on the manifold of orthogonal matrices and associated Procrustes approximations of the TSP cost function. Using these dynamical systems, we analyze the local neighborhood around the optimal TSP solutions (which are equilibria) using computations to approximate the associated stable manifolds. We find that these flows frequently converge to undesirable equilibria. However, the solutions of the dynamical systems and the associated Procrustes approximation provide an interesting biasing approach for the popular Lin–Kernighan heuristic which yields fast convergence. The Lin–Kernighan heuristic is typically based on the computation of edges that have a “high probability” of being in the shortest tour, thereby effectively pruning the search space. Our new approach, instead, relies on a natural relaxation of the combinatorial optimization problem to the manifold of orthogonal matrices and the subsequent use of this solution to bias the Lin–Kernighan heuristic. Although the initial cost of computing these edges using the Procrustes solution is higher than existing methods, we find that the Procrustes solution, when coupled with a homotopy computation, contains valuable information regarding the optimal edges. We explore the Procrustes-based approach on several TSP instances and find that our approach on average requires fewer k-opt moves than existing approaches. Broadly, this work connects dynamical systems theory with combinatorial optimization to provide algorithmic and computational complexity insights.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others


  1. The trace of a matrix \( A \in \mathbb {R}^{n \times n} \) is defined to be the sum of all diagonal entries, i.e., \( {{\,\mathrm{tr}\,}}(A) = \sum _{i=1}^n a_{ii} \).


  1. Agarwala, R., Applegate, D.L., Maglott, D., Schuler, G.D., Schäffer, A.A.: A fast and scalable radiation hybrid map construction and integration strategy. Genome Res. 10(3), 350–364 (2000)

    Google Scholar 

  2. Applegate, D., Bixby, R., Cook, W., Chvátal, V.: On the solution of traveling salesman problems. Rheinische Friedrich-Wilhelms-Universität Bonn (1998)

  3. Anstreicher, K., Wolkowicz, H.: On Lagrangian relaxation of quadratic matrix constraints. SIAM J. Matrix Anal. Appl. 22(1), 41–55 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Blackford, L.S., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., et al.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  5. Burkard, R.E., Çela, E., Pardalos, P.M., Pitsoulis, L.S.: The quadratic assignment problem. In: Pardalos, P.M., Du, D.-Z. (eds.) Handbook of Combinatorial Optimization, pp. 1713–1809. Springer, Berlin (1998)

    Google Scholar 

  6. Biere, A., Heule, M., van Maaren, H.: Handbook of Satisfiability, vol. 185. IOS Press, Amsterdam (2009)

    MATH  Google Scholar 

  7. Brockett, R.W.: Least squares matching problems. Linear Algebra Appl. 122–124, 761–777 (1989)

    MathSciNet  MATH  Google Scholar 

  8. Brockett, R.W.: Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems. Linear Algebra Appl. 146, 79–91 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Carlson, S.: Algorithm of the gods. Sci. Am. 276, 121–124 (1997)

    Google Scholar 

  10. Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, DTIC Document (1976)

  11. Chung, F.: Spectral Graph Theory, vol. 92. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  12. Cook, W.J.: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation. Princeton University Press, Princeton (2011)

    Google Scholar 

  13. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT press (2009)

  14. Dellnitz, M., Froyland, G., Junge, O.: The algorithms behind GAIO—set oriented numerical methods for dynamical systems. In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 145–174. Springer, Berlin (2001)

  15. Dorigo, M., Gambardella, L.M.: Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans. Evol. Comput. 1(1), 53–66 (1997)

    Google Scholar 

  16. Dellnitz, M., Hohmann, A.: The computation of unstable manifolds using subdivision and continuation. In: Nonlinear Dynamical Systems and Chaos, pp. 449–459. Springer, Berlin (1996)

  17. Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75, 293–317 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Ercsey-Ravasz, M., Toroczkai, Z.: Optimization hardness as transient chaos in an analog approach to constraint satisfaction. Nat. Phys. 7(12), 966 (2011)

    Google Scholar 

  20. Englot, B., Sahai, T., Cohen, I.: Efficient tracking and pursuit of moving targets by heuristic solution of the traveling salesman problem. In: 52nd IEEE Conference on Decision and Control, pp. 3433–3438. IEEE (2013)

  21. Finke, G., Burkard, R.E., Rendl, F.: Quadratic assignment problems. North-Holland Math. Stud. 132, 61–82 (1987)

    MathSciNet  MATH  Google Scholar 

  22. Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(2), 298–305 (1973)

    MathSciNet  MATH  Google Scholar 

  23. Fiedler, M.: Laplacian of graphs and algebraic connectivity. Banach Center Publ. 25(1), 57–70 (1989)

    MathSciNet  MATH  Google Scholar 

  24. Gower, J.C., Dijksterhuis, G.B.: Procrustes Problems, vol. 3. Oxford University Press, New York (2004)

    MATH  Google Scholar 

  25. Grötschel, M., Jünger, M., Reinelt, G.: Optimal control of plotting and drilling machines: a case study. Math. Methods Oper. Res. 35(1), 61–84 (1991)

    MATH  Google Scholar 

  26. Goldschmidt, O., Laugier, A., Olinick, E.V.: SONET/SDH ring assignment with capacity constraints. Discrete Appl. Math. 129(1), 99–128 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Helsgaun, K.: An effective implementation of the Lin–Kernighan traveling salesman heuristic. Datalogiske Skrifter (Writings on Computer Science), No. 81, Roskilde University (1998)

  28. Helsgaun, K.: An effective implementation of K-opt moves for the Lin-Kernighan TSP heuristic. PhD thesis, Roskilde University. Department of Computer Science (2006)

  29. Helsgaun, K.: General k-opt submoves for the Lin–Kernighan TSP heuristic. Math. Program. Comput. 1(2–3), 119–163 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Helsgaun, K.: Solving the bottleneck traveling salesman problem using the Lin–Kernighan–Helsgaun algorithm. Technical Report, Computer Science, Roskilde University (2014)

  31. Helsgaun, K.: Solving the equality generalized traveling salesman problem using the Lin–Kernighan–Helsgaun algorithm. Computer Science Report, 141 (2014)

  32. Higham, N.J.: Computing the polar decomposition—with applications. SIAM J. Sci. Stat. Comput. 7(4), 1160–1174 (1986)

    MathSciNet  MATH  Google Scholar 

  33. Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18(6), 1138–1162 (1970)

    MathSciNet  MATH  Google Scholar 

  34. Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees: part II. Math. Program. 1, 6–25 (1971)

    MATH  Google Scholar 

  35. Hunt, B.R., Kaloshin, V.Y.: Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity 12(5), 1263–1275 (1999)

    MathSciNet  MATH  Google Scholar 

  36. Hadley, S.W., Rendl, F., Wolkowicz, H.: Bounds for the quadratic assignment problems using continuous optimization techniques. In IPCO, pp. 237–248 (1990)

  37. Hadley, S.W., Rendl, F., Wolkowicz, H.: A new lower bound via projection for the quadratic assignment problem. Math. Oper. Res. 17, 727–739 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Karp, R.M.: Reducibility among combinatorial problems. In: 50 Years of Integer Programming 1958–2008, pp. 219–241 (2010)

  39. Koopmans, T.C., Beckmann, M.: Assignment problems and the location of economic activities. Econometrica 25(1), 53–76 (1957)

    MathSciNet  MATH  Google Scholar 

  40. Kolemen, E., Kasdin, N.J.: Optimal trajectory control of an occulter-based planet-finding telescope. American Astronautical Society, pp. 07–037 (2007)

  41. Klus, S., Sahai, T.: A spectral assignment approach for the graph isomorphism problem. Inf. Inference J. IMA 7(4), 689–706 (2018)

    Google Scholar 

  42. Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling-salesman problem. Oper. Res. 21(2), 498–516 (1973)

    MathSciNet  MATH  Google Scholar 

  43. Lenstra, J.K., Kan, A.H.G.: Complexity of vehicle routing and scheduling problems. Networks 11(2), 221–227 (1981)

    Google Scholar 

  44. Platt, J.C., Barr, A.H.: Constrained differential optimization for neural networks. Technical report, California Institute of Technology (1988)

  45. Petersen, K.B., Pedersen, M.S.: The matrix cookbook (2008)

  46. Padberg, M., Rinaldi, G.: A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Rev. 33(1), 60–100 (1991)

    MathSciNet  MATH  Google Scholar 

  47. Reinelt, G.: TSPLIB—a traveling salesman problem library. ORSA J. Comput. 3(4), 376–384 (1991)

    MATH  Google Scholar 

  48. Su, W., Boyd, S., Candes, E.: A differential equation for modeling Nesterov’s accelerated gradient method: theory and insights. In Advances in Neural Information Processing Systems, pp. 2510–2518 (2014)

  49. Schönemann, P.: On two-sided orthogonal Procrustes problems. Psychometrika 33(1), 19–33 (1968)

    MathSciNet  Google Scholar 

  50. Sahai, T., Klus, S.: Automatic learning of Bayesian networks, May. US Patent App. 14/546,392 (2015)

  51. Sahai, T., Speranzon, A., Banaszuk, A.: Wave equation based algorithm for distributed eigenvector computation. In 2010 49th IEEE Conference on Decision and Control (CDC), pp. 7308–7315. IEEE (2010)

  52. Sahai, T., Speranzon, A., Banaszuk, A.: Hearing the clusters of a graph: a distributed algorithm. Automatica 48(1), 15–24 (2012)

    MathSciNet  MATH  Google Scholar 

  53. Schäfer, F., Sullivan, T.J., Owhadi, H.: Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity. arXiv preprint arXiv:1706.02205 (2017)

  54. Stewart, W.R.: Accelerated branch exchange heuristics for symmetric traveling salesman problems. Networks 17(4), 423–437 (1987)

    MathSciNet  MATH  Google Scholar 

  55. Sahai, T., Ziessler, A., Klus, S., Dellnitz, M.: Continuous relaxations for the traveling salesman problem (2017). arXiv preprint arXiv:1702.05224

  56. Wong, W.S.: Matrix representation and gradient flows for NP-hard problems. J. Optim. Theory Appl. 87, 197–220 (1995)

    MathSciNet  MATH  Google Scholar 

  57. Wibisono, A., Wilson, A.C., Jordan, M.I.: A variational perspective on accelerated methods in optimization. Proc. Natl. Acad. Sci. 113(47), E7351–E7358 (2016)

    MathSciNet  MATH  Google Scholar 

  58. Wen, Z., Yin, W.: A feasible method for optimization with orthogonality constraints. Technical report, Rice University (2010)

  59. Ziessler, A., Dellnitz, M., Gerlach, R.: The numerical computation of unstable manifolds for infinite dimensional dynamical systems by embedding techniques. Submitted to SIAM J. Appl. Dyn. Syst. (2018). arXiv:1808.08787

  60. Zavlanos, M.M., Pappas, G.: A dynamical systems approach to weighted graph matching. Automatica 44, 2817–2824 (2008)

    MathSciNet  MATH  Google Scholar 

Download references


The authors thank Prof. Keld Helsgaun for discussions related to the Lin–Kernighan heuristic and his software and also Dr. Mirko Hessel-von Molo and Steffen Ridderbusch for discussions related to the approach. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) and Space and Naval Warfare Systems Center Pacific (SSC Pacific) under Contract No. N6600118C4031.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Tuhin Sahai.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sahai, T., Ziessler, A., Klus, S. et al. Continuous relaxations for the traveling salesman problem. Nonlinear Dyn 97, 2003–2022 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: