Minimum Vehicle Routing with a Common Deadline

Abstract

In this paper, we study the following vehicle routing problem: given n vertices in a metric space, a specified root vertex r (the depot), and a length bound D, find a minimum cardinality set of r-paths that covers all vertices, such that each path has length at most D. This problem is $\mathcal{NP}$-complete, even when the underlying metric is induced by a weighted star. We present a 4-approximation for this problem on tree metrics. On general metrics, we obtain an O(logD) approximation algorithm, and also an $(O(\log \frac{1}{\epsilon}),1+\epsilon)$ bicriteria approximation. All these algorithms have running times that are almost linear in the input size. On instances that have an optimal solution with one r-path, we show how to obtain in polynomial time, a solution using at most 14 r-paths.

We also consider a linear relaxation for this problem that can be solved approximately using techniques of Carr & Vempala [7]. We obtain upper bounds on the integrality gap of this relaxation both in tree metrics and in general.