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The travelling salesman and the PQ-tree

Part of the Lecture Notes in Computer Science book series (LNCS,volume 1084)

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.

Keywords

  • Travelling salesman problem
  • Polynomial algorithm
  • Dynamic programming
  • Combinatorial optimization
  • Euclidean travelling salesman problem
  • PQ-tree

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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© 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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