Improved Approximations for the Unsplittable Capacitated Vehicle Routing Problem

2026-01-29

AI Summary

This study addresses the uncapacitated capacitated vehicle routing problem (UCVRP) with indivisible demands, where each customer must be served entirely by a single route, and the objective is to minimize total travel cost. The authors propose novel approximation algorithms for both fixed and general vehicle capacity settings. By integrating the current best-known metric TSP approximation ratio α, they design a route partitioning and merging strategy based on logarithmic optimization and constant-root computations. In the fixed-capacity case, the algorithm achieves an approximation ratio better than 3.0897; for general capacities, it improves upon the previous bound of 3.1932 with a ratio below 3.1759. These results not only surpass the longstanding 3.1932 barrier but also admit further improvement as advances in TSP approximation continue.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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).

Problem

• Unsplittable Capacitated Vehicle Routing Problem
• Vehicle Routing
• Approximation Algorithms
• Combinatorial Optimization
• Metric TSP

Innovation

• Unsplittable CVRP
• Approximation Algorithm
• Vehicle Routing Problem
• Metric TSP
• Combinatorial Optimization