Estimating the Held-Karp lower bound for the geometric TSP

European Journal of Operational Research

Volumn 102, Issue 1, 1997, Pages 157-175

  Valenzuela, Christine L ^a   Jones, Antonia J ^b  

^a TEESSIDE UNIVERSITY   (United Kingdom)

^b CARDIFF UNIVERSITY   (United Kingdom)

Author keywords

Held Karp lower bound; Lagrangian relaxation; Minimum 1 tree; Subgradient optimization; Travelling salesman

Indexed keywords

ESTIMATION; ITERATIVE METHODS; LAGRANGE MULTIPLIERS; LINEAR PROGRAMMING; PROBLEM SOLVING;

HELD KARP LOWER BOUND (HK); SUBGRAPH SPEED UP TECHNIQUE; TRAVELLING SALESMAN PROBLEM (TSP);

OPERATIONS RESEARCH;

EID: 0031246770     PISSN: 03772217     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-2217(96)00214-7     Document Type: Article

References (31)

1. Applegate, D., Bixby, R., Chvátal, V. and Cook, W., 1995. Finding cuts in the TSP (A preliminary report). Report No. 95-05, Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), Rutgers University, Piscataway, N.J.
2. Balas, E., and Toth, P., 1985. Branch and Bound Methods. In: E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys (eds), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley and Sons, New York.
3. Beardwood, J., Halton J.H. and Hammersley, J.M., 1959. The shortest path through many points. Proceedings of the Cambridge Philosophical Society, 55, 299-327.
4. Bently, J.L., 1975. Multidimensional binary search trees used for associative search. Communications of the ACM, 18, 309-517.
5. Christophides, N., 1979. The Traveling Salesman Problem. In: N. Christophides, A. Mingozzi, P. Toth and C. Sandi (eds), Combinatorial Optimization. John Wiley and Sons, New York.
6. Goemans, M.X. and Bertsimas, D.J., 1991. Probabilistic Analysis of the Held and Karp Lower Bound for the Euclidean Traveling Salesman Problem. Mathematics of Operations Research, 16(1), 72-89.
7. Helbig-Hansen, K.H. and Krarup, J., 1974. Improvements of the Held-Karp algorithm for the symmetric traveling salesman problem. Mathematical Programming, 7, 87-96.
8. Held, M. and Karp, R.M., 1970. The Traveling Salesman Problem and Minimum Spanning Trees. Operations Research, 18, 1138-1162.
9. Held, M. and Karp, R.M., 1971. The Traveling Salesman Problem and Minimum Spanning Trees, part II. Mathematical Programming, 1, 6-25.
10. Held, M., Wolfe, P. and Crowder, H.P., 1974. Validation of subgradient optimization. Mathematical Programming, 6, 62-88.
11. Horowitz, E. and Sahni, S., 1978. Fundamentals of Computer Algorithms, Pitman.
12. Johnson, D.S., 1988. Talk presented at the Mathematical Programming Symposium, Tokyo.
13. Johnson, D.S., 1990. Local Optimization and the Traveling Salesman Problem. Automata Languages and Programming, 17th International Colloquium Proceedings.
14. Johnson, D.S. and Mcgeoch, L.A., 1995a. The Traveling Salesman Problem: A Case Study in Local Optimization. To appear in: E.H.L. Aarts and J.K. Lenstra (eds), Local Search in Combinatoric Optimization, John Wiley and Sons, New York.
15. Johnson, D.S., 1995b. Personal communication.
16. Johnson, D.S., McGeoch, L.A. and Rothberg, E.E., 1996. Asymptotic experimental analysis for the Held-Karp traveling salesman bound. Proceedings of the 1996 ACM-SIAM symposium on Discrete Algorithms.
17. Polyak, B.T., 1967. A general method of solving extremal problems. Soviet Mathematics Doklady, 8, 593-597. (Translation of Doklady Akademii Nauk SSSR 174, 1967.)
18. Polyak, B.T., 1967. A general method of solving extremal problems. Soviet Mathematics Doklady, 8, 593-597. (Translation of Doklady Akademii Nauk SSSR 174, 1967.)
19. Polyak, B.T., 1969. Minimization of unsmooth functional. Computational Mathematics and Mathematical Physics 14-29. (Translation of Žurnal Vyčislitel'noǐ Matematiki i Matematičesko Fiziki, 9, 509-521.)
20. Polyak, B.T., 1969. Minimization of unsmooth functional. Computational Mathematics and Mathematical Physics 14-29. (Translation of Žurnal Vyčislitel'noǐ Matematiki i Matematičesko Fiziki, 9, 509-521.)
21. Reinelt, G., 1994. The Traveling Salesman: Computational Solutions for TSP Applications. Lecture Notes in Computer Science, 840, Springer-Verlag, Berlin.
22. Reinelt, G., 1995. A library of TSP problems from: E-mail: elib@ZIB-Berlin.de. Telnet: elib.zib-berlin.de. Just send the message 'help' via email to find out how to use elib.
23. Shamos, M.I. and Hoey, D., 1975. Closest Point Problems. Proceedings of the 16th IEEE Annual Symposium on Foundations of Computer Science, 151-162.
24. Shmoys, D.B. and Williamson, D.P., 1990. Analyzing the Held-Karp TSP Bound: A Monotonicity Property with Application. Information Processing Letters, 35, 281-285.
25. Stein, D., 1977. Scheduling Dial-a-Ride transportation systems: an asymptotic approach. Ph.D thesis, Harvard University, Cambridge, MA.
26. Valenzuela, C.L. and Jones, A.J., 1994. Evolutionary Divide and Conquer(I): a novel genetic approach to the TSP. Evolutionary Computation, 1(4), 313-333.
27. Valenzuela, C.L. and Jones, A.J., 1995a. A Parallel Implementation of Evolutionary Divide and Conquer for the TSP. Proceedings of the First IEE/ IEEE conference on Genetic ALgorithms in Engineering Systems: Innovations and Applications (GALESIA) Sheffield U.K. 12-14 September 1995, 499-504.
28. Valenzuela, C.L., 1995b. Evolutionary Divide and Conquer: a novel genetic approach to the TSP. Unpublished Ph.D. thesis, Imperial College, University of London.
29. Volgenant, T. and Jonker, R., 1982. A branch and bound algorithm for the symmetric traveling salesman problem based on the 1-tree relaxation. European Journal of Operational Research, 9, 83-89.
30. Wichmann, B.A. and Hill, D., 1982. A Pseudo-random Number Generator. National Physical Laboratory U.K. Report DITC 6-82.
31. Wolsey, L., 1980. Heuristic analysis, linear programming and branch and bound. Mathematical Programming Study, 13, 121-134.