Faster algorithms for orienteering and k-TSP

We consider the rooted orienteering problem in Euclidean space: Given n points P R d , a root point s ∈ and budget B > 0 find path that starts from has total length at most visits as many of possible. This is known to be NP-hard, hence we study ( 1 − δ ) -approximation algorithms. The previous Polynomial-Time Approximation Scheme (PTAS) for this problem, due Chen Har-Peled (2008), runs time O / 2 improving on bound was left an open problem. Our main contribution PTAS with significantly improved complexity . A technique approximating reduce it solving correlated instances k -TSP (a tour one least points). However, tours reduction must achieve certain excess guarantee (namely, their can surpass optimum only proportion parameter called excess) stronger than usual + -approximation. technical improve running these variants, particularly its dependence dimension Indeed, our polynomial even moderately large dimension, roughly up = log ⁡ instead