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 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 n O(n) + O(n 3 log ∑ c k c ) and requires n ᶿ(n) space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its logarithmic dependence on the total length ∑ c k c of the tour, allowing the algorithm to handle instances with very long tours. The superexponential dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this article, we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time 2 O(n) , i.e., single-exponential in the number of cities, using polynomial space. The space requirement of our algorithm is (essentially) the size of the output, and assuming the Exponential-Time Hypothesis (ETH), the problem cannot be solved in time 2 o(n) . Our algorithm is deterministic, and arguably both simpler and easier to analyze than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.

中文翻译：

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多次访问 TSP 的时空最优算法

多次访问旅行商问题 (MV-TSP)n访问每个城市的城市C规定的数目ķC 次。旅行成本可能是不对称的，连续两次访问一个城市可能会产生非零成本。MV-TSP 问题在某些图族的调度、几何近似和哈密顿性方面有应用。已知最快的 MV-TSP 算法归功于 Cosmadakis 和 Papadimitriou (SICOMP, 1984)。它在时间 n 中运行在）+ O(n3对数∑CķC) 并且需要 nᶿ(n)空间。Cosmadakis-Papadimitriou 算法的一个有趣的特点是它对数依赖于总长度 ∑ CķC 游览，允许算法处理具有很长游览的实例。这超指数然而，在时间和空间复杂度上对城市数量的依赖使得该算法对于除此参数的最窄范围之外的所有对象都是不切实际的。在本文中，我们改进了 Cosmadakis-Papadimitriou 算法，给出了一个在时间 2 中运行的 MV-TSP 算法在）， IE，单指数在城市数量中，使用多项式空间。我们算法的空间要求（本质上）是输出的大小，并且假设指数时间假设（ETH），问题不能在时间 2 中解决在）. 我们的算法是确定性的，可以说比 Cosmadakis 和 Papadimitriou 的原始方法更简单、更容易分析。它涉及对定向生成树的优化和递归的、基于质心的树分解。