TY - GEN

T1 - An approximation algorithm for a bottleneck traveling salesman problem

AU - Kao, Ming Yang

AU - Sanghi, Manan

N1 - Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2006

Y1 - 2006

N2 - Consider a truck running along a road. It picks up a load Li at point βi and delivers it at αi, carrying at most one load at a time. The speed on the various parts of the road in one direction is given by f(x) and that in the other direction is given by g(x). Minimizing the total time spent to deliver loads L1,...,Ln is equivalent to solving the Traveling Salesman Problem (TSP) where the cities correspond to the loads Li with coordinates (αi, βi) and the distance from Li to Lj is given by ∫αiβj f(x)dx if βj ≥ αi and by ∫βjαi g(x)dx if βj < αi. This case of TSP is polynomially solvable with significant real-world applications. Gilmore and Gomory obtained a polynomial time solution for this TSP [6]. However, the bottleneck version of the problem (BTSP) was left open. Recently, Vairaktarakis showed that BTSP with this distance metric is NP-complete [10]. We provide an approximation algorithm for this BTSP by exploiting the underlying geometry in a novel fashion. This also allows for an alternate analysis of Gilmore and Gomory's polynomial time algorithm for the TSP. We achieve an approximation ratio of (2 + γ) where γ ≥ f(x)/g(x) ≥ 1/γ ∀x. Note that when f(x) = g(x), the approximation ratio is 3.

AB - Consider a truck running along a road. It picks up a load Li at point βi and delivers it at αi, carrying at most one load at a time. The speed on the various parts of the road in one direction is given by f(x) and that in the other direction is given by g(x). Minimizing the total time spent to deliver loads L1,...,Ln is equivalent to solving the Traveling Salesman Problem (TSP) where the cities correspond to the loads Li with coordinates (αi, βi) and the distance from Li to Lj is given by ∫αiβj f(x)dx if βj ≥ αi and by ∫βjαi g(x)dx if βj < αi. This case of TSP is polynomially solvable with significant real-world applications. Gilmore and Gomory obtained a polynomial time solution for this TSP [6]. However, the bottleneck version of the problem (BTSP) was left open. Recently, Vairaktarakis showed that BTSP with this distance metric is NP-complete [10]. We provide an approximation algorithm for this BTSP by exploiting the underlying geometry in a novel fashion. This also allows for an alternate analysis of Gilmore and Gomory's polynomial time algorithm for the TSP. We achieve an approximation ratio of (2 + γ) where γ ≥ f(x)/g(x) ≥ 1/γ ∀x. Note that when f(x) = g(x), the approximation ratio is 3.

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U2 - 10.1007/11758471_23

DO - 10.1007/11758471_23

M3 - Conference contribution

AN - SCOPUS:33746078429

SN - 354034375X

SN - 9783540343752

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 223

EP - 235

BT - Algorithms and Complexity - 6th Italian Conference, CIAC 2006, Proceedings

PB - Springer Verlag

T2 - 6th Italian Conference on Algorithms and Complexity, CIAC 2006

Y2 - 29 May 2006 through 31 May 2006

ER -