# The directed orienteering problem

## 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(log 2n/log log n. logk) -approximation algorithm for this problem. We use this algorithm for directed k-TSP to obtain an O(log 2n/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. © 2011 Springer Science+Business Media, LLC.