Abstract
This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed k-TSP problem: given an asymmetric metric (V,d), a root r∈V and a target k≤|V|, compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time \(O(\frac{\log^{2} n}{\log\log n}\cdot\log k)\)-approximation algorithm for this problem. We use this algorithm for directed k-TSP to obtain an \(O(\frac{\log^{2} n}{\log\log n})\)-approximation algorithm for the directed orienteering problem. This answers positively, the question of poly-logarithmic approximability of directed orienteering, an open problem from Blum et al. (SIAM J. Comput. 37(2):653–670, 2007). The previously best known results were quasi-polynomial time algorithms with approximation guarantees of O(log 2 k) for directed k-TSP, and O(log n) for directed orienteering (Chekuri and Pal in IEEE Symposium on Foundations in Computer Science, pp. 245–253, 2005). Using the algorithm for directed orienteering within the framework of Blum et al. (SIAM J. Comput. 37(2):653–670, 2007) and Bansal et al. (ACM Symposium on Theory of Computing, pp. 166–174, 2004), we also obtain poly-logarithmic approximation algorithms for the directed versions of discounted-reward TSP and vehicle routing problem with time-windows.
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A preliminary version of this paper appeared as [21].
Work by V. Nagarajan was done while at CMU and supported in part by NSF grants ITR CCR-0122581 (The ALADDIN project) and CCF-0728841.
R. Ravi was supported in part by NSF grants CCF-0430751 and ITR grant CCR-0122581 (The ALADDIN project).
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Nagarajan, V., Ravi, R. The Directed Orienteering Problem. Algorithmica 60, 1017–1030 (2011). https://doi.org/10.1007/s00453-011-9509-2
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DOI: https://doi.org/10.1007/s00453-011-9509-2