Probabilistic Analysis of Euclidean Capacitated Vehicle Routing

TL;DR

This paper provides a probabilistic analysis of the Euclidean capacitated vehicle routing problem with random customer locations, improving bounds on the approximation ratio of the iterated tour partitioning algorithm.

Contribution

It introduces a new lower bound on the optimal routing cost and refines the approximation ratio bounds for the ITP algorithm in the random setting.

Findings

ITP algorithm is at best a (1+c_0)-approximation when k=√n.

Improved upper bound on ITP approximation ratio to 0.915+α.

New lower bound on the optimal cost for the problem.

Abstract

We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of $n$ unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most $k$ customers, where $k$ is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely. The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when $k$ is either $o(\sqrt{n})$ or $\omega(\sqrt{n})$, and they asked whether the ITP algorithm was also effective in the intermediate range. In this work, we show that when $k=\sqrt{n}$, the ITP algorithm is at best a $(1+c_0)$-approximation for some positive constant $c_0$. On the other hand, the approximation ratio of the ITP algorithm was known to be at most $0.995+\alpha$ due to Bompadre, Dror, and Orlin, where $\alpha$ is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to $0.915+\alpha$. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest.