Abstract
Let D = (dij) be the n x n distance matrix of a set of n cities {1,2,..., n}, and let T be a PQ-tree with node degree bounded by d that represents a set II(T) of permutations over {1, 2,..., n}. We show how to compute for D in O(2d n 3) time the shortest travelling salesman tour contained in II(T). Our algorithm may be interpreted as a common generalization of the well-known Held and Karp dynamic programming algorithm for the TSP and of the dynamic programming algorithm for finding the shortest pyramidal TSP tour.
This result has two surprising consequences. The first consequence concerns large sets of permutations, so-called exponential neighborhoods, over which the TSP can be solved efficiently. Up to now, the largest known, neighborhoods had cardinality \(2^{\Theta (n)}\), whereas our result yields new neighborhoods of cardinality \(2^{\Theta (n\log \log n)}\). The second consequence is that the shortcutting phase of the “twice around the tree” heuristic for the Euclidean TSP can be optimally implemented in polynomial time. This contradicts a statement of Papadimitriou and Vazirani as published in 1984.
This research has been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.
Supported by a research fellowship of the Euler Institute for Discrete Mathematics and its Applications.
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References
F. Aurenhammer, On-line sorting of twisted sequences in linear time, BIT 28, 1988, 194–204.
K.S. Booth and G.S. Lueker, Testing for the consecutive ones property, interval graphs and graph planarity using PQ-tree algorithms, Journal of Computer and System Sciences 13, 1976, 335–379.
R.E. Burkard and J.A.A. Van der Veen, Universal conditions for algebraic traveling salesman problems to be efficiently solvable, Optimization 22, 1991, 787–814.
J. Carlier and P. Villon, A new heuristic for the travelling salesman problem, KAIRO — Operations Research 24, 1990, 245–253.
N. Christofides, Worst-Case Analysis of a new heuristic for the travelling salesman problem, Technical Report CMU, 1976.
G.A. Croes, A method for solving travelling-salesman problems, Operations Research 6, 1958, 791–812.
P.C. Gilmore, E.L. Lawler and D.B. Shmoys, Well-solved special cases, Chapter 4 in [12], 87–143.
F. Glover and A.P. Punnen, The travelling salesman problem: New solvable cases and linkages with the development of approximation algorithms, Technical report, University of Colorado, Boulder, 1995.
M. Held and R.M. Karp, A dynamic programming approach to sequencing problems, J. SIAM 10, 1962, 196–210.
D.S. Johnson and C.H. Papadimitriou, Performance guarantees for heuristics, Chapter 5 in [12], 145–180.
P.S. Klyaus, The structure of the optimal solution of certain classes of travelling salesman problems, (in Russian), Vestsi Akad. Nauk USSR, Physics and Math., Sci., Minsk, 1976, 95–98.
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, The travelling salesman problem, Wiley, Chichester, 1985.
S. Lin, Computer solutions to the travelling salesman problem, Bell System Tech. J. 44, 1965, 2245–2269.
C.H. Papadimitriou, The Euclidean travelling salesman problem is NP-complete, Theoretical Computer Science 4, 1977, 237–244.
C.H. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity, Prentice Hall, New Jersey, 1982.
C.H. Papadimitriou and U.V. Vazirani, On two geometric problems related to the travelling salesman problem, J. Algorithms 5, 1984, 231–246.
C.N. Potts and S.L. van de Velde, Dynasearch — Iterative local improvement by dynamic programming: Part I, The travelling salesman problem, Technical Report, University of Twente, The Netherlands, 1995.
A. Serdyukov, On some extremal walks in graphs, Uprarfiaemye systemy 17, Institute of Mathematics of the Siberian Academy of Sciences, USSR, 1978, 76–79.
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© 1996 Springer-Verlag Berlin Heidelberg
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Burkard, R.E., Deineko, V.G., Woeginger, G.J. (1996). The travelling salesman and the PQ-tree. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_36
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DOI: https://doi.org/10.1007/3-540-61310-2_36
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