TY - JOUR

T1 - Faster algorithms for orienteering and k-TSP

AU - Gottlieb, Lee Ad

AU - Krauthgamer, Robert

AU - Rika, Havana

N1 - Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/5/7

Y1 - 2022/5/7

N2 - We consider the rooted orienteering problem in Euclidean space: Given n points P in Rd, 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 nO(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 nO(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(loglogn) instead of d=O(1).

AB - We consider the rooted orienteering problem in Euclidean space: Given n points P in Rd, 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 nO(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 nO(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(loglogn) instead of d=O(1).

KW - Orienteering

KW - Plane sweep algorithm

KW - k-TSP

UR - http://www.scopus.com/inward/record.url?scp=85126126090&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2022.02.013

DO - 10.1016/j.tcs.2022.02.013

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AN - SCOPUS:85126126090

SN - 0304-3975

VL - 914

SP - 73

EP - 83

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -