An approximation algorithm for a bottleneck traveling salesman problem

Abstract

Consider a truck running along a road. It picks up a load L i at point β i and delivers it at α i , 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 L 1 , … , L n is equivalent to solving the traveling salesman problem (TSP) where the cities correspond to the loads L i with coordinates ( α i , β i ) and the distance from L i to L j is given by ∫ α i β j f ( x ) d x if β j ⩾ α i and by ∫ β j α i g ( x ) d x if β j < α i . Gilmore and Gomory obtained a polynomial time solution for this TSP [P.C. Gilmore, R.E. Gomory, Sequencing a one state-variable machine: A solvable case of the traveling salesman problem, Operations Research 12 (1964) 655–679]. However, the bottleneck version of the problem (BTSP) was left open. Recently, Vairaktarakis showed that BTSP with this distance metric is NP-complete [G.L. Vairaktarakis, On Gilmore–Gomory's open question for the bottleneck TSP, Operations Research Letters 31 (2003) 483–491]. We provide an approximation algorithm for this BTSP by exploiting the underlying geometry in a novel fashion. 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.