Abstract
We consider two natural generalizations of the Asymmetric Traveling Salesman problem: the k-Stroll and the k-Tour problems. The input to the k-Stroll problem is a directed n-vertex graph with nonnegative edge lengths, an integer k, as well as two special vertices s and t. The goal is to find a minimum-length s-t walk, containing at least k distinct vertices (including the endpoints s,t). The k-Tour problem can be viewed as a special case of k-Stroll, where s=t. That is, the walk is required to be a tour, containing some pre-specified vertex s. When k=n, the k-Stroll problem becomes equivalent to Asymmetric Traveling Salesman Path, and k-Tour to Asymmetric Traveling Salesman.
Our main result is a polylogarithmic approximation algorithm for the k-Stroll problem. Prior to our work, only bicriteria (O(log2 k),3)-approximation algorithms have been known, producing walks whose length is bounded by 3OPT, while the number of vertices visited is Ω(k/log2 k). We also show a simple O(log2 n/loglogn)-approximation algorithm for the k-Tour problem. The best previously known approximation algorithms achieved min(O(log3 k),O(log2 n⋅logk/loglogn)) approximation in polynomial time, and O(log2 k) approximation in quasipolynomial time.
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Notes
k-Tour is sometimes referred to as k-ATSP in the literature. Similarly, k-Stroll is sometimes called k-ATSPP.
Since we will be focusing on directed graphs, the names k-Tour and k-Stroll will refer to the directed versions of the problems throughout the paper, unless stated otherwise.
In fact, the solution is always a path with k vertices since we assume the graph is complete and its edge lengths satisfy triangle inequality.
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Acknowledgements
We would like to thank Chandra Chekuri for suggesting the problems, and for sharing with us his survey on open problems related to Orienteering.
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M. Bateni was supported in part by a Gordon Wu fellowship, a Charlotte Elizabeth Procter fellowship as well as NSF ITR grants CCF-0205594, CCF-0426582 and NSF CCF 0832797, NSF CAREER award CCF-0237113, MSPA-MCS award 0528414, NSF expeditions award 0832797. Work was done while the author was a graduate student in Princeton University, and an intern in Toyota Technological Institute.
J. Chuzhoy supported in part by NSF CAREER award CCF-0844872.
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Bateni, M., Chuzhoy, J. Approximation Algorithms for the Directed k-Tour and k-Stroll Problems. Algorithmica 65, 545–561 (2013). https://doi.org/10.1007/s00453-011-9610-6
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DOI: https://doi.org/10.1007/s00453-011-9610-6