A time- And space-optimal algorithm for the many-visits TSP

The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of n cities that visits each city c a prescribed number kc 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 Pc kc) and requires nO(n) space. The interesting feature of the Cosmadakis-Papadimitriou algorithm is its logarithmic dependence on the total length Pc kc of the tour, allowing the algorithm to handle instances with very long tours, beyond what is tractable in the standard TSP setting. However, its superexponential dependence on the number of cities in both its time and space complexity renders the algorithm impractical for all but the narrowest range of this parameter. In this paper we significantly improve on the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time 2O(n), i.e. single-exponential in the number of cities, with polynomial space. The space requirement of our algorithm is (essentially) the size of the output, and assuming the Exponential-time Hypothesis (ETH), the time requirement is optimal. 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.