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 $\int^{\beta_j}_{\alpha_i} f(x)dx$ if β _j ≥ α _i and by $\int^{\alpha_i}_{\beta_j} g(x)dx$ if β _j < α _i . 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 $\gamma \geq \frac{f(x)}{g(x)} \geq \frac{1}{\gamma} \; \forall x$. Note that when f(x)=g(x), the approximation ratio is 3.

Supported in part by NSF Grant EIA-0112934.