Many visits TSP revisited
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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 O^*(5^n). They also show a polynomial space algorithm running in time O^*(16^n+o(n)). In this work, we show three main results: (i) A randomized polynomial space algorithm in time O^*(2^nD), where D is the maximum distance between two cities. By using standard methods, this results in (1+ϵ)-approximation in time O^*(2^nϵ^-1). Improving the constant 2 in these results would be a major breakthrough, as it would result in improving the O^*(2^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 O^*(4^n) running time bound. (iii) A new polynomial space algorithm, running in time O(7.88^n).
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