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

Wed, May 29, 2019 12:30 CEST

Speakers: Matthias Mnich
Tags: Algorithms, Travelling Salesman Problem

The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of n cities that visits each city c a prescribed number kcof 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 nO(n)+O(n3log∑ckc) and requires nΩ(n) space.

The interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length ∑ckc 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.

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 2O(n)+O(n3log∑ckc), with O(n2log∑ckc) 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.

Appeared at SODA 2019, joint work with André Berger, Laszló Kozma and Roland Vincze.