# The travelling salesman and the PQ-tree

## 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*(2^{d}*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## Preview

Unable to display preview. Download preview PDF.

## References

- 1.F. Aurenhammer, On-line sorting of twisted sequences in linear time,
*BIT***28**, 1988, 194–204.Google Scholar - 2.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.Google Scholar - 3.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.Google Scholar - 4.J. Carlier and P. Villon, A new heuristic for the travelling salesman problem,
*KAIRO — Operations Research***24**, 1990, 245–253.Google Scholar - 5.N. Christofides, Worst-Case Analysis of a new heuristic for the travelling salesman problem, Technical Report CMU, 1976.Google Scholar
- 6.G.A. Croes, A method for solving travelling-salesman problems,
*Operations Research***6**, 1958, 791–812.Google Scholar - 7.P.C. Gilmore, E.L. Lawler and D.B. Shmoys, Well-solved special cases, Chapter 4 in [12], 87–143.Google Scholar
- 8.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.Google Scholar
- 9.M. Held and R.M. Karp, A dynamic programming approach to sequencing problems,
*J. SIAM***10**, 1962, 196–210.Google Scholar - 10.D.S. Johnson and C.H. Papadimitriou, Performance guarantees for heuristics, Chapter 5 in [12], 145–180.Google Scholar
- 11.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.Google Scholar - 12.E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys,
*The travelling salesman problem*, Wiley, Chichester, 1985.Google Scholar - 13.S. Lin, Computer solutions to the travelling salesman problem, Bell System Tech. J.
**44**, 1965, 2245–2269.Google Scholar - 14.C.H. Papadimitriou, The Euclidean travelling salesman problem is NP-complete, Theoretical Computer Science
**4**, 1977, 237–244.CrossRefGoogle Scholar - 15.C.H. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity, Prentice Hall, New Jersey, 1982.Google Scholar
- 16.C.H. Papadimitriou and U.V. Vazirani, On two geometric problems related to the travelling salesman problem, J. Algorithms
**5**, 1984, 231–246.CrossRefGoogle Scholar - 17.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.Google Scholar
- 18.A. Serdyukov, On some extremal walks in graphs,
*Uprarfiaemye systemy***17**, Institute of Mathematics of the Siberian Academy of Sciences, USSR, 1978, 76–79.Google Scholar