TSP Heuristics: Domination Analysis and Complexity

We show that the 2-Opt and 3-Opt heuristics for the traveling salesman problem (TSP) on the complete graph Kn produce a solution no worse than the average cost of a tour in Kn in a polynomial number of iterations. As a consequence, we get that the domination numbers of the 2- Opt , 3- Opt , Carlier—Villon, Shortest Path Ejection Chain, and Lin—Kernighan heuristics are all at least (n-2)! / 2 . The domination number of the Christofides heuristic is shown to be no more than $\lceil{n}/{2}\rceil !$ , and for the Double Tree heuristic and a variation of the Christofides heuristic the domination numbers are shown to be one (even if the edge costs satisfy the triangle inequality). Further, unless P = NP, no polynomial time approximation algorithm exists for the TSP on the complete digraph $\vec{K}_n$ with domination number at least (n-1)!-k for any constant k or with domination number at least (n-1)! - (( k /(k+1))(n+r))!-1 for any non-negative constants r and k such that (n+r) $\equiv$ 0 mod (k+1). The complexities of finding the median value of costs of all the tours in $\vec{K}_n$ and of similar problems are also studied.

Authors and Affiliations

1. Department of Mathematical Sciences, University of New Brunswick, Saint John, New Brunswick, Canada E2L 4L5. punnen@unbsj.ca., Canada

   Punnen

2. Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA. fmargot@ms.uky.edu., USA

   Margot

3. Faculty of Administration, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A4. kabadi@unb.ca., Canada

   Kabadi

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