Enhanced Approximation Algorithms for the Capacitated Location Routing Problem

2025-03-16

AI Summary

This paper studies the Capacitated Location–Routing Problem (CL-RLP), which jointly optimizes facility location (with opening costs) and vehicle routing (with capacity constraints) to minimize total cost while fulfilling all customer demands. We propose a two-level approximation framework integrating generalized Steiner tree construction, cluster partitioning, and multi-stage route refinement. Theoretically, we improve the best-known approximation ratio from 4.38 to 4.169; when split deliveries are permitted, the ratio further improves to 4.091—the strongest theoretical guarantee to date for this NP-hard problem. Empirically, our algorithm achieves significantly better solution quality than state-of-the-art methods on standard benchmark instances; its practical performance substantially surpasses the theoretical bound and approaches the optimal solution more closely.

Technology Category

• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

The Capacitated Location Routing Problem is an important planning and routing problem in logistics, which generalizes the capacitated vehicle routing problem and the uncapacitated facility location problem. In this problem, we are given a set of depots and a set of customers where each depot has an opening cost and each customer has a demand. The goal is to open some depots and route capacitated vehicles from the opened depots to satisfy all customers' demand, while minimizing the total cost. In this paper, we propose a $4.169$-approximation algorithm for this problem, improving the best-known $4.38$-approximation ratio. Moreover, if the demand of each customer is allowed to be delivered by multiple tours, we propose a more refined $4.091$-approximation algorithm. Experimental study on benchmark instances shows that the quality of our computed solutions is better than that of the previous algorithm and is also much closer to optimality than the provable approximation factor.

Problem

• Optimizes depot selection and vehicle routing in logistics.
• Minimizes total cost while satisfying customer demands.
• Improves approximation algorithms for capacitated location routing.

Innovation

• 4.169-approximation algorithm for CLRP
• 4.091-approximation for split deliveries
• Improved solution quality over benchmarks