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 $k_c$ of times. Travel costs may not be symmetric, 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 $n^{O(n)} + O(n^3 \log \sum_c k_c )$ and requires $n^{\Omega(n)}$ space. The interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length $\sum_c k_c$ of the tour, allowing the algorithm to handle instances with very long tours, beyond what is tractable in the standard TSP setting. However, the \emph{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 \emph{single-exponential} time with \emph{polynomial} space. More precisely, we obtain the running time $2^{O(n)} + O(n^3 \log{\sum_c k_c})$, with $O(n^2 \log{\sum_c k_c})$ space. Assuming the Exponential-time Hypothesis (ETH), both the time and space requirements of our algorithm are 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 oriented spanning trees of a graph and employs a recursive, centroid-based decomposition of trees.