Faster algorithms for orienteering and k-TSP

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(dd/δ)2^(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)2^(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(log⁡log⁡n) instead of d=O(1).