Faster Algorithms for Orienteering and k-TSP
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We consider the rooted orienteering problem in Euclidean space: Given n points P in ℝ^d, a root point s∈ P and a budget ℬ>0, find a path that starts from s, has total length at most ℬ, and visits as many points of P as possible. This problem is known to be NP-hard, hence we study (1-δ)-approximation algorithms. The previous Polynomial-Time Approximation Scheme (PTAS) for this problem, due to Chen and Har-Peled (2008), runs in time n^O(d√(d)/δ)(log n)^(d/δ)^O(d), and improving on this time bound was left as an open problem. Our main contribution is a PTAS with a significantly improved time complexity of n^O(1/δ)(log n)^(d/δ)^O(d). A known technique for approximating the orienteering problem is to reduce it to solving 1/δ correlated instances of rooted k-TSP (a k-TSP tour is one that visits at least k points). However, the k-TSP tours in this reduction must achieve a certain excess guarantee (namely, their length can surpass the optimum length only in proportion to a parameter of the optimum called excess) that is stronger than the usual (1+δ)-approximation. Our main technical contribution is to improve the running time of these k-TSP variants, particularly in its dependence on the dimension d. Indeed, our running time is polynomial even for a moderately large dimension, roughly up to d=O(loglog n) instead of d=O(1).
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