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Annals of Operations Research

, Volume 86, Issue 0, pp 529–558 | Cite as

New classes of efficiently solvable generalized Traveling Salesman Problems

  • E. Balas
Article

Abstract

We consider the n‐city traveling salesman problem (TSP), symmetric or asymmetric,with the following attributes. In one case, a positive integer k and an ordering (1,..., n) ofthe cities is given, and an optimal tour is sought subject to the condition that for any pairi, j ∈ (1..., n), if j ≥ i + k, then i precedes j in the tour. In another case, position i in the tourhas to be assigned to some city within k positions from i in the above ordering. This case isclosely related to the TSP with time windows. In a third case, an optimal tour visiting m outof n cities is sought subject to constraints of the above two types. This is a special case ofthe Prize Collecting TSP (PCTSP). In any of the three cases, k may be replaced by city‐specificintegers k(i), i = 1,..., n. These problems arise in practice. For each class, we reducethe problem to that of finding a shortest source‐sink path in a layered network with a numberof arcs linear in n and exponential in the parameter k (which is independent of the problemsize). Besides providing linear time algorithms for the solution of these problems, the reductionto a shortest path problem also provides a compact linear programming formulation.Finally, for TSPs or PCTSPs that do not have the required attributes, these algorithms canbe used as heuristics that find in linear time a local optimum over an exponential‐sizeneighborhood.

Keywords

Positive Integer Time Window Short Path Local Optimum Linear Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    N. Ascheuer, L.F. Escudero, M. Grötschel and M. Stoer, On identifying in polynomial time violated subtour elimination and precedence forcing constraints for the sequential ordering problem, in: Integer Programming and Combinatorial Optimization, (Proceedings of IPCO 1), eds. R. Kannan and W.R. Pulleyblank, University of Waterloo Press, 1990, pp. 19-28.Google Scholar
  2. [2]
    E. Balas, The Prize Collecting Traveling Salesman Problem, ORSA/TIMS Meeting, Los Angeles, Spring 1986.Google Scholar
  3. [3]
    E. Balas, The Prize Collecting Traveling Salesman Problem, Networks 19(1989)621-636.Google Scholar
  4. [4]
    E. Balas, The Prize Collecting Traveling Salesman Problem: II. Polyhedral results, Networks 25(1995)199-216.Google Scholar
  5. [5]
    E. Balas, M. Fischetti and W.R. Pulleyblank, The precedence constrained asymmetric Traveling Salesman Polytope, Mathematical Programming 68(1995)241-265.Google Scholar
  6. [6]
    E. Balas and C.H. Martin, Combinatorial optimization in steel rolling (Extended Abstract), Workshop on Combinatorial Optimization in Science and Technology (COST), RUTCOR, Rutgers University, April 1991.Google Scholar
  7. [7]
    E. Balas and N. Simonetti, Linear time dynamic programming algorithms for new classes of restricted TSPs: A computational study, MSRR No. 625, GSIA, Carnegie Mellon University, October 1997.Google Scholar
  8. [8]
    X. Berenguer, A characterization of linear admissible transformations for the m-Traveling Salesman Problem, European Journal of Operations Research 3(1979)232-249.Google Scholar
  9. [9]
    V.Ya. Burdyuk and V.N. Trofimov, Generalization of the results of Gilmore and Gomory on the solution of the Traveling Salesman Problem (in Russian), Izv. Akad. Nauk SSSR, Tech. Kibernet, 3(1976)16-22 [English translation in Engineering Cybernetics 14(1976)12-18].Google Scholar
  10. [10]
    R.E. Burkard and W. Sandholzer, Efficiently solvable special cases of the Bottleneck Traveling Salesman Problem, Discrete Applied Mathematics 32(1991)61-67.Google Scholar
  11. [11]
    R.E. Burkard and J.A.A. van der Veen, Universal conditions for algebraic Travelling Salesman Problems to be efficiently solvable, Optimization 22(1991)787-814.Google Scholar
  12. [12]
    R.E. Burkard and V.G. Deineko, Polynomially solvable cases of the Traveling Salesman Problem and a new exponential neighborhood, Computing 54(1995)191-211.Google Scholar
  13. [13]
    R. van Dal, J.A.A. van der Veen and G. Sierksma, Small and large TSP: Two well-solvable cases of the Traveling Salesman Problem, Eur. J. of Oper. Res. 69(1993)107-120.Google Scholar
  14. [14]
    G. Cornuéjols, D. Naddef and W.R. Pulleyblank, Halin graphs and the Traveling Salesman Problem, Mathematical Programming 26(1983)287-294.Google Scholar
  15. [15]
    V.M. Demidenko, The Traveling Salesman Problem with asymmetric matrices (in Russian), Vestsi Ak. Navuk BSSR, Ser. Fiz.-Mat. Navuk 1(1979)29-35.Google Scholar
  16. [16]
    L. Escudero, M. Guidnard and K. Malik, On identifying and lifting valid cuts for the sequential ordering problem with precedence relationships and deadlines, University of Madrid and IBM Spain, 1991.Google Scholar
  17. [17]
    W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, Wiley, 1957.Google Scholar
  18. [18]
    M.T. Fiala and W.R. Pulleyblank, Precedence in constrained routing and helicopter scheduling: Heuristic design, Interfaces 22(1992)100-111.Google Scholar
  19. [19]
    M. Fischetti and P. Toth, An additive approach for the optimal solution of the Prize Collecting Traveling Salesman Problem, in: Vehicle Routing: Methods and Studies, eds. B.L. Golden and A.A. Assad, North-Holland, 1988, pp. 319-343.Google Scholar
  20. [20]
    S. Fuller, An optimal drum scheduling algorithm, IEEE Trans. Comput. C-21(1972)1153-1165.Google Scholar
  21. [21]
    E.Ya. Gabovich, The small Traveling Salesman Problem (in Russian), Trudy Vychisl Tsentra Tartu, Gos. Univ. 19(1970)27-51.Google Scholar
  22. [22]
    E.Ya. Gabovich, Constant discrete programming problems on substitution sets (in Russian), Kibernetika 5(1976)128-134 [translation: Cybernetics 12(1977)786-793].Google Scholar
  23. [23]
    N.E. Gaikov, On the minimization of a linear form on cycles (in Russian), Vestsi Ak. Navuk BSSR, Ser. Fiz.-Mat. Navuk 4(1980)128.Google Scholar
  24. [24]
    R.S. Garfinkel, Minimizing wallpaper waste: A class of Traveling Salesman Problems, Operations Research 25(1977)741-751.Google Scholar
  25. [25]
    P.C. Gilmore and R.E. Gomory, Sequencing a one state variable machine: A solvable case of the Traveling Salesman Problem, Operations Research 12(1964)655-679.Google Scholar
  26. [26]
    P.C. Gilmore, E.L. Lawler and D. Shmoys, Well-solved special cases, in: The Traveling Salesman Problem: A Guided Tour to Combinatorial Optimization, eds. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D. Shmoys, Wiley, 1985, chap. 4.Google Scholar
  27. [27]
    K. Kalmanson, Edgeconvex circuits and the Traveling Salesman Problem, Canadian Journal of Mathematics 27(1975)1000-1010.Google Scholar
  28. [28]
    P.S. Klyaus, Structure of the optimal solution of certain classes of Traveling Salesman Problems (in Russian), Vestsi Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk 6(1976)95-98.Google Scholar
  29. [29]
    E.L. Lawler, A solvable case of the Traveling Salesman Problem, Mathematical Programming 1(1971)267-269.Google Scholar
  30. [30]
    J.K. Park, A special case of the n-vertex Traveling Salesman Problem that can be solved in O(n) time, Inform. Process. Lett. 40(1991)247-254.Google Scholar
  31. [31]
    H.D. Ratliff and A.S. Rosenthal, Order picking in a rectangular warehouse: A solvable case of the Traveling Salesman Problem, Operations Research 31(1983)507-521.Google Scholar
  32. [32]
    N. Simonetti and E. Balas, Implementation of a linear time algorithm for certain generalized Traveling Salesman Problems, in: Integer Programming and Combinatorial Optimization (Proceedings of IPCO~5), eds. W.H. Cunningham, T.S. McCormick and M. Queyranne, Springer, 1996, pp. 316-329.Google Scholar
  33. [33]
    F. Supnick, Extreme Hamiltonian lines, Annals of Mathematics 65(1957)179-201.Google Scholar
  34. [34]
    J.A.A. van der Veen, Solvable cases of the Travelling Salesman Problem with various objective functions, Ph.D. Thesis, Nijenrode University, 1992.Google Scholar
  35. [35]
    J.A.A. van der Veen, G. Sierksma, R. van Dal, Pyramidal tours and the Travelling Salesman Problem, European Journal of Operational Research 69(1993)107-120.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

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  • E. Balas

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