Probabilistic Analysis of Partitioning Algorithms for the Traveling-Salesman Problem in the Plane

We consider partitioning algorithms for the approximate solution of large instances of the traveling-salesman problem in the plane. These algorithms subdivide the set of cities into small groups, construct an optimum tour through each group, and then patch the subtours together to form a tour through all the cities. If the number of cities in the problem is n, and the number of cities in each group is t, then the worst-case error is $O\sqrt{n/t}$ . If the cities are randomly distributed, then the relative error is Ot-1/2 with probability one. Hybrid schemes are suggested, in which partitioning is used in conjunction with existing heuristic algorithms. These hybrid schemes may be expected to give near-optimum solutions to problems with thousands of cities.

[1]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  M. Held,et al.  A dynamic programming approach to sequencing problems , 1962, ACM National Meeting.

[3]  Richard Bellman,et al.  Dynamic Programming Treatment of the Travelling Salesman Problem , 1962, JACM.

[4]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[5]  Patrick D. Krolak,et al.  A man-machine approach toward solving the traveling salesman problem , 1970, CACM.

[6]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[7]  Patrick D. Krolak,et al.  A man-machine approach toward solving the traveling salesman problem , 1970, DAC '70.

[8]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[9]  D. J. Rosenkrantz,et al.  Approximate Algorithms for the Traveling Salesperson Problem , 1974, SWAT.

[10]  Jakob Krarup,et al.  Improvements of the Held—Karp algorithm for the symmetric traveling-salesman problem , 1974, Math. Program..

[11]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[12]  Michael Ian Shamos,et al.  Geometric complexity , 1975, STOC.

[13]  Ronald L. Graham,et al.  Some NP-complete geometric problems , 1976, STOC '76.

[14]  Christos H. Papadimitriou,et al.  The Euclidean Traveling Salesman Problem is NP-Complete , 1977, Theor. Comput. Sci..

[15]  Nicos Christofides.  Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem , 1976, Operations Research Forum.