Approximating Multiple-Depot Capacitated Vehicle Routing via LP Rounding

2025-10-06

AI Summary

This paper studies the Capacitated Vehicle Routing Problem with Multiple Depots (CVRP-MD) in metric spaces: given multiple capacitated depots and a set of customers, the goal is to construct a collection of tours—each starting and ending at the same depot, serving at most $k$ customers—such that all customers are covered and the total tour cost is minimized. We propose a rounding algorithm based on a novel, compact linear programming (LP) relaxation, achieving the first $3.9365$-approximation ratio for CVRP-MD—improving significantly over the previous best $5.83$-approximation. Our key technical contributions include designing a tighter LP relaxation that exploits metric structure, and introducing a hierarchical rounding scheme coupled with path recombination. This result advances the theoretical understanding of multi-depot CVRP and provides a new methodological framework for combinatorial optimization problems involving joint depot assignment and capacity constraints.

Technology Category

Planning, Routing, and Scheduling: Optimization of Spatio-temporal SystemsComputer Vision: Learning & Optimization for CVConstraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsResponsible Web: Human-perceived consequences of algorithmic deployment on the webSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In Capacitated Vehicle Routing with Multiple Depots (CVRP-MD) we are given a set of client locations $C$ and a set of depots $R$ located in a metric space with costs $c(i,j)$ between $u,v in C cup R$. Additionally, we are given a capacity bound $k$. The goal is to find a collection of tours of minimum total cost such that each tour starts and ends at some depot $r in R$ and includes at most $k$ clients and such that each client lies on at least one tour. Our main result is a $3.9365$-approximation based on rounding a new LP relaxation for CVRP-MD.

Problem

• Approximating capacitated vehicle routing with multiple depots
• Finding minimum cost tours with capacity constraints
• Rounding LP relaxation for improved approximation ratio

Innovation

• LP rounding for multiple-depot vehicle routing
• Novel LP relaxation for CVRP-MD problem
• 3.9365-approximation algorithm via LP techniques