TSP tour domination and Hamilton cycle decompositions of regular digraphs
Section snippets
Introduction, terminology and notation
It is well known that most combinatorial optimization problems are -hard. Due to the lack of polynomial time algorithms to solve -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 , 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 .
- 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 For every n⩾2, n≠4, n≠6, there exists a decomposition of into tours. Lemma 2.2 For a positive integer k<n, a k-Tillson [21]
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 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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