TSP tour domination and Hamilton cycle decompositions of regular digraphs

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Abstract

In this paper, we solve a problem by Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502–510) from the context of domination analysis, where the performance of a heuristic algorithm is rated by the number of solutions that are not better than the solution found by the algorithm, rather than by the relative performance compared to the optimal value. In particular, we show that for the asymmetric traveling salesman problem, there is a deterministic polynomial time algorithm that finds a tour that is at least as good as the median of all tour values. Our algorithm uses an unpublished theorem by Häggkvist on the Hamilton decomposition of regular digraphs.

Section snippets

Introduction, terminology and notation

It is well known that most combinatorial optimization problems are NP-hard. Due to the lack of polynomial time algorithms to solve NP-hard problems to optimality, the following two approaches to deal with such problems have been developed. The first one is the design of polynomial approximation algorithms that produce feasible solutions whose value is always within a constant factor of the optimum. Unfortunately, many important combinatorial optimization problems, including the asymmetric

Results

We start with a brief description of our algorithm applied to (Kn,c), where n is large enough.

  • 1.

    Compute k such that every k-regular digraph of order n has a decomposition into Hamilton cycles (see Theorem 2.3 for details).

  • 2.

    Find a minimum cost k-regular spanning subgraph M of (Kn,c).

  • 3.

    Find the minimum cost tour Z in a Hamilton cycle decomposition of M. Return Z.

Now we shall study this algorithm.

Lemma 2.1

Tillson [21]

For every n⩾2, n≠4, n≠6, there exists a decomposition of Kn into tours.

Lemma 2.2

For a positive integer k<n, a k-

Remarks and further research

It is worth noting that Häggkvist's decomposition result cannot be improved, in a sense, since the digraph of order n with two connected components isomorphic to Kn/2 is an (n/2−1)-regular digraph. This means that to improve the 50% threshold, another approach is needed. It would be very interesting to have a solution to the Glover–Punnen problem with p being constant, which does not rely heavily on previous results.

The algorithm suggested in this paper appears to be impractical due to the

Acknowledgments

We are very grateful to the referee, whose remarks and suggestions allowed us to significantly improve the presentation of this note.

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