Papers › A time- and space-optimal algorithm for the many-visits TSP
A time- and space-optimal algorithm for the many-visits TSP
André Berger, László Kozma, Matthias Mnich, Roland Vincze
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The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of n cities that visits each city c a prescribed number k_c of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time nᴼ⁽ⁿ⁾ + O(n³ log∑_c k_c ) and requires n^(Θ(n)) space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length ∑_c k_c of the tour, allowing the algorithm to handle instances with very long tours. The \emph{superexponential} dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time 2ᴼ⁽ⁿ⁾, i.e.\ \emph{single-exponential} in the number of cities, using \emph{polynomial} space. Our algorithm is deterministic, and arguably both simpler and easier to analyse than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.
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