A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees

TL;DR

This paper presents a 4/3-approximation algorithm for the Capacitated Vehicle Routing problem on trees, matching the known lower bound and thus establishing the best possible approximation ratio.

Contribution

It introduces a tight 4/3-approximation algorithm for capacitated vehicle routing on trees, improving previous ratios and matching the lower bound for optimality.

Findings

Achieves a 4/3 approximation ratio for the problem.

Proves the approximation ratio is tight and cannot be improved.

Improves upon the previous best-known approximation ratio.

Abstract

Given a set of clients with demands, the Capacitated Vehicle Routing problem is to find a set of tours that collectively cover all client demand, such that the capacity of each vehicle is not exceeded and such that the sum of the tour lengths is minimized. In this paper, we provide a 4/3-approximation algorithm for Capacitated Vehicle Routing on trees, improving over the previous best-known approximation ratio of $(\sqrt{41}-1)/4$ by Asano et al., while using the same lower bound. Asano et al. show that there exist instances whose optimal cost is 4/3 times this lower bound. Notably, our 4/3 approximation ratio is therefore tight for this lower bound, achieving the best-possible performance.