The Asymmetric Traveling Salesman Problem
Abstract
Starting with M(a), an n X n asymmetric cost matrix, Jonker and Volgenannt transformed it into a 2n X 2n symmetric cost matrix, M(s)where M(s) has unusual properties. One such property is that an optimal tour in M(s) yields an optimal tour in M(a). Modifying M(s), we apply the modified Floyd-Warshall algorithm to M(s). Due to the structure of M(s), we hopefully)can always obtain an optimal tour in M(a) in polynomial time.If theorem 1 in this paper is valid, since the asymmetric traveling salesman problem is NP-hard, P would equal NP.
Keywords
Cite
@article{arxiv.math/0612114,
title = {The Asymmetric Traveling Salesman Problem},
author = {Howard Kleiman},
journal= {arXiv preprint arXiv:math/0612114},
year = {2007}
}
Comments
PDF. WORD 2000, 2 pages.The following error in the proof of theorem 1 has been corrected: "a negatively-valued cycle in M(s)" has been replaced by "a minimal positively-valued cycle in M(s)". Also, "ATPS" should have been "ATSP". Also, the running time of the algorithm should be "at most O(n^4) since, when extending a path, we often may have to backtrack to insure that a path stays acceptable. Finally, assuming that the proof is valid, since the asymmetric problem is NP-hard, P would equal NP