An Approximation Algorithm for Distance-Constrained Vehicle Routing on Trees

TL;DR

This paper presents a simple approximation algorithm for the distance-constrained vehicle routing problem on trees, achieving a ratio of approximately 1.691, improving previous bounds, with a novel analysis based on online bin packing reduction.

Contribution

It introduces a natural approximation algorithm for tree DVRP with a tight analysis using a new reduced length concept, improving the approximation ratio from 2 to about 1.691.

Findings

Approximation ratio of about 1.691 for tree DVRP

Algorithm's analysis is tight and based on bin packing reduction

Polynomial runtime in number of terminals and distance constraint

Abstract

In the Distance-constrained Vehicle Routing Problem (DVRP), we are given a graph with integer edge weights, a depot, a set of $n$ terminals, and a distance constraint $D$. The goal is to find a minimum number of tours starting and ending at the depot such that those tours together cover all the terminals and the length of each tour is at most $D$. The DVRP on trees is of independent interest, because it is equivalent to the virtual machine packing problem on trees studied by Sindelar et al. [SPAA'11]. We design a simple and natural approximation algorithm for the tree DVRP, parameterized by $\varepsilon >0$. We show that its approximation ratio is $\alpha + \varepsilon$, where $\alpha \approx 1.691$, and in addition, that our analysis is essentially tight. The running time is polynomial in $n$ and $D$. The approximation ratio improves on the ratio of 2 due to Nagarajan and Ravi [Networks'12]. The main novelty of this paper lies in the analysis of the algorithm. It relies on a reduction from the tree DVRP to the bounded space online bin packing problem via a new notion of reduced length.