Abstract
In this paper, we introduce the notion of l-quasi-pyramidal and l-pseudo-pyramidal tours extending the classic notion of pyramidal tours to the case of the Generalized Traveling Salesman Problem (GTSP). We show that, for the instance of GTSP on n cities and k clusters with arbitrary weights, l-quasi-pyramidal and l-pseudo-pyramidal optimal tours can be found in time \(O(4^ln^3)\) and \(O(2^lk^{l+4}n^3)\), respectively. Consequently, we show that, in the most general setting, GTSP belongs to FPT for parametrizations induced by these special kinds of tours. Also, we describe a non-trivial polynomially solvable subclass of GTSP, for which the existence of l-quasi-pyramidal optimal tour (for some fixed value of l) is proved.
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Notes
- 1.
In this paper, we restrict ourselves to the case of undirected graphs. Nevertheless, the analogous argument can be provided to the case of digraphs and asymmetric weighting functions w.
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This research was supported by Russian Science Foundation, project no. 14-11-00109.
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Khachay, M., Neznakhina, K. (2017). Generalized Pyramidal Tours for the Generalized Traveling Salesman Problem. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10627. Springer, Cham. https://doi.org/10.1007/978-3-319-71150-8_23
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