Improved Approximations for the Unsplittable Capacitated Vehicle Routing Problem

TL;DR

This paper introduces two improved approximation algorithms for the unplittable capacitated vehicle routing problem, reducing the approximation ratio closer to the best-known bounds by leveraging advanced mathematical techniques.

Contribution

The authors develop two new approximation algorithms for the CVRP, one for fixed capacity and one for general capacity, with improved ratios over previous methods.

Findings

Approximation ratio for fixed capacity is less than 3.0897.

Approximation ratio for general capacity is less than 3.1759.

Both algorithms can be further refined using recent mathematical results.

Abstract

The capacitated vehicle routing problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. In this problem, we are given a depot and a set of customers, each with a demand, embedded in a metric space. The objective is to find a set of tours, each starting and ending at the depot, operated by the capacititated vehicle at the depot to serve all customers, such that all customers are served, and the total travel cost is minimized. We consider the unplittable variant, where the demand of each customer must be served entirely by a single tour. Let $\alpha$ denote the current best-known approximation ratio for the metric traveling salesman problem. The previous best approximation ratio was $\alpha+1+\ln 2+\delta<3.1932$ for a small constant $\delta>0$ (Friggstad et al., Math. Oper. Res. 2025), which can be further improved by a small constant using the result of Blauth, Traub, and Vygen (Math. Program. 2023). In this paper, we propose two improved approximation algorithms. The first algorithm focuses on the case of fixed vehicle capacity and achieves an approximation ratio of $\alpha+1+\ln\bigl(2-\frac{1}{2}y_0\bigr)<3.0897$, where $y_0>0.39312$ is the unique root of $\ln\bigl(2-\frac{1}{2}y\bigr)=\frac{3}{2}y$. The second algorithm considers general vehicle capacity and achieves an approximation ratio of $\alpha+1+y_1+\ln\left(2-2y_1\right)+\delta<3.1759$ for a small constant $\delta>0$, where $y_1>0.17458$ is the unique root of $\frac{1}{2} y_1+ 6 (1-y_1)\bigl(1-e^{-\frac{1}{2} y_1}\bigr) =\ln\left(2-2y_1\right)$. Both approximations can be further improved by a small constant using the result of Blauth, Traub, and Vygen (Math. Program. 2023).