Many visits TSP revisited

Abstract: We study the Many Visits TSP problem, where given a number $k(v)$ for each of $n$ cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city $v$ exactly $k(v)$ times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020] and runs in time and space $\mathcal{O}<sup>*(5<sup>n)$. They also show a polynomial space algorithm running in time $\mathcal{O}<sup>*(16<sup>{n+o(n)})$. In this work, we show three main results: (i) A randomized polynomial space algorithm in time $\mathcal{O}<sup>*(2<sup>nD)$, where $D$ is the maximum distance between two cities. By using standard methods, this results in $(1+\epsilon)$-approximation in time $\mathcal{O}<sup>*(2<sup>n\epsilon<sup>{-1})$. Improving the constant $2$ in these results would be a major breakthrough, as it would result in improving the $\mathcal{O}<sup>*(2<sup>n)$-time algorithm for Directed Hamiltonian Cycle, which is a 50 years old open problem. (ii) A tight analysis of Berger et al.'s exponential space algorithm, resulting in $\mathcal{O}<sup>*(4<sup>n)$ running time bound. (iii) A new polynomial space algorithm, running in time $\mathcal{O}(7.88<sup>n)$.