Improvements of the Held—Karp algorithm for the symmetric traveling-salesman problem
A highly efficient algorithm (HK) devised by Held and Karp for solving the symmetric traveling-salesman problem was presented at the 7th Mathematical Programming Symposium in 1970 and published in Mathematical Programming in 1971. Its outstanding performance is due to a clever exploitation of the relationship between the traveling-salesman problem and minimum spanning trees.
However, various improvements of their method have led to a version (IHK) which tends to be some 25 times faster than the original one. Experiments with data selected at random, ranging in size up to 80 cities, show that the computing time for IHK is roughly doubled as the number of cities is increased by 10.
KeywordsComputing Time Mathematical Method Span Tree Mathematical Program Efficient Algorithm
Unable to display preview. Download preview PDF.
- E.W. Dijkstra, “A note on two problems in connexion with graphs”,Numerische Mathematik 1 (1959) 269–271.Google Scholar
- M.H. van Emden, “Increasing the efficiency of quicksort”, Algorithm 402,Communications of the Association for Computing Machinery 13 (1970) 693–695.Google Scholar
- K. Helbig Hansen, “Sekvensproblemer med vaegt på modifikation af Held og Karp's algoritme”, M.Sc. thesis, Institute of Datalogy, University of Copenhagen, Denmark (1971).Google Scholar
- M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees”,Operations Research 18 (1970) 1138–1162.Google Scholar
- M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees: Part II”,Mathematical Programming 1 (1971) 6–25.Google Scholar
- J.B. Kruskal, “On the shortest spanning subtree of a graph and the traveling-salesman problem”,Proceedings of the American Mathematical Society 2 (1956) 48–50.Google Scholar
- R.C. Prim, “Shortest connection networks and some generalizations”,Bell System Technical Journal 36 (1957) 1389–1401.Google Scholar