Abstract
We consider the s-t-path TSP: given a finite metric space with two elements s and t, we look for a path from s to t that contains all the elements and has minimum total distance. We improve the approximation ratio for this problem from 1.599 to 1.566. Like previous algorithms, we solve the natural LP relaxation and represent an optimum solution \(x^*\) as a convex combination of spanning trees. Gao showed that there exists a spanning tree in the support of \(x^*\) that has only one edge in each narrow cut (i.e., each cut C with \(x^*(C)<2\)). Our main theorem says that the spanning trees in the convex combination can be chosen such that many of them are such “Gao trees” simultaneously at all sufficiently narrow cuts.
This work was done during the trimester program on combinatorial optimization at the Hausdorff Institute for Mathematics in Bonn.
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Acknowledgement
We thank Kanstantsin Pashkovich for allowing us to include his idea described in the second half of Sect. 3.
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Gottschalk, C., Vygen, J. (2016). Better s-t-Tours by Gao Trees. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_11
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