Abstract
Optimization heuristics are often compared with each other to determine which one performs best by means of worst-case performance ratio reflecting the quality of returned solution in the worst case. The domination number is a complement parameter indicating the quality of the heuristic in hand by determining how many feasible solutions are dominated by the heuristic solution. We prove that the Max-Regret heuristic introduced by Balas and Saltzman (Oper. Res. 39:150–161, 1991) finds the unique worst possible solution for some instances of the s-dimensional (s≥3) assignment and asymmetric traveling salesman problems of each possible size. We show that the Triple Interchange heuristic (for s=3) also introduced by Balas and Saltzman and two new heuristics (Part and Recursive Opt Matching) have factorial domination numbers for the s-dimensional (s≥3) assignment problem.
Article PDF
Similar content being viewed by others
References
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Berlin (1999)
Balas, E., Saltzman, M.J.: An algorithm for the three-index assignment problem. Oper. Res. 39, 150–161 (1991)
Bendall, G., Margot, F.: Greedy type resistance of combinatorial problems. Discret. Optim. 3, 288–298 (2006)
Berend, D., Skiena, S., Twitto, Y.: Combinatorial dominance guarantees for problems with infeasible solutions (2007, submitted).
Burkard, R., Cela, E.: Linear assignment problems and extensions. In: Du, Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, pp. 75–149. Kluwer Academic, Dordrecht (1999)
Ghosh, D., Goldengorin, B., Gutin, G., Jäger, G.: Tolerance-based greedy algorithms for the traveling salesman problem. Communic. DQM 10, 52–70 (2007)
Glover, F., Punnen, A.: The traveling salesman problem: new solvable cases and linkages with the development of approximation algorithms. J. Oper. Res. Soc. 48, 502–510 (1997)
Glover, F., Gutin, G., Yeo, A., Zverovich, A.: Construction heuristics for the asymmetric TSP. Eur. J. Oper. Res. 129, 555–568 (2001)
Goldengorin, B., Jäger, G., Molitor, P.: Tolerances applied in combinatorial optimization. J. Comput. Sci. 2(9), 716–734 (2006)
Gupta, P., Kahng, A.B., Mantik, S.: Routing-aware scan chain ordering. ACM Trans. Des. Autom. Electron. Syst. 10(3), 546–560 (2005)
Gutin, G., Punnen, A.: The Traveling Salesman Problem and its Variations. Kluwer Academic, Dordrecht (2002)
Gutin, G., Yeo, A.: Domination analysis of combinatorial optimization algorithms and problems. In: Golumbic, M., Hartman, I. (eds.) Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications. Springer, Berlin (2005)
Gutin, G., Yeo, A.: Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number. Discret. Appl. Math. 119, 107–116 (2002)
Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126, 106–130 (2000)
Johnson, D.S., Gutin, G., McGeoch, L., Yeo, A., Zhang, X., Zverovitch, A.: Experimental analysis of heuristics for ATSP. In: Gutin, G., Punnen, A. (eds.) The Traveling Salesman Problem and its Variations. Kluwer Academic, Dordrecht (2002)
Koller, A.E., Noble, S.D.: Domination analysis of greedy heuristics for the frequency assignment problem. Discret. Math. 275, 331–338 (2004)
Murphey, R., Pardalos, P.M., Pitsoulis, L.S.: A parallel GRASP for the data association multidimensional assignment problem. In: Parallel Processing of Discrete Problems. The IMA Volumes in Mathematics and its Applications, vol. 106, pp. 159–180 (1998)
Pierskalla, W.P.: The multidimensional assignment problem. Oper. Res. 16, 422–431 (1968)
Poore, A.B.: Multidimensional assignment formulation of data association problems arising from multitarget and multisensor tracking. Comput. Optim. Appl. 3, 27–54 (1994)
Punnen, A.P.: The traveling salesman problem: applications, formulations and variations. In: Gutin, G., Punnen, A. (eds.) The Traveling Salesman Problem and its Variations. Kluwer Academic, Dordrecht (2002)
Punnen, A.P., Kabadi, S.: Domination analysis of some heuristics for the traveling salesman problem. Discret. Appl. Math. 119, 117–128 (2002)
Punnen, A.P., Margot, F., Kabadi, S.N.: TSP heuristics: domination analysis and complexity. Algorithmica 35, 111–127 (2003)
Pusztaszeri, J., Rensing, P.E., Liebling, T.M.: Tracking elementary particles near their primary vertex: a combinatorial approach. J. Glob. Optim. 16, 422–431 (1995)
Reinfeld, N.V., Vogel, W.R.: Mathematical Programming. Prentice-Hall, Englewood Cliffs (1958)
Robertson, A.J.: A set of greedy randomized adaptive local search procedure implementations for the multidimensional assignment problem. Comput. Optim. Appl. 19, 145–164 (2001)
Xu, L., Tan, A.C., Naiman, D.Q., Geman, D., Winslow, R.L.: Robust prostate cancer marker genes emerge from direct integration of inter-study microarray data. Bioinformatics 21, 3905–3911 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gutin, G., Goldengorin, B. & Huang, J. Worst case analysis of Max-Regret, Greedy and other heuristics for Multidimensional Assignment and Traveling Salesman Problems. J Heuristics 14, 169–181 (2008). https://doi.org/10.1007/s10732-007-9033-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-007-9033-3