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 K _n produce a solution no worse than the average cost of a tour in K_nin 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 , 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 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) 0 mod (k+1). The complexities of finding the median value of costs of all the tours in and of similar problems are also studied.