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Enhanced Approximation Algorithms for the Capacitated Location Routing Problem

Jingyang Zhao, Mingyu Xiao, Shunwang Wang

TL;DR

This work advances the Capacitated Location Routing Problem (CLR) by introducing two approximation schemes. Tree-Alg achieves a $4.169$-approximation for both unsplittable and splittable CLR, while Path-Alg attains a refined $4.091$-approximation for splittable CLR, each supported by novel lower bounds and reductions to UFL. The authors develop an α-parameterized framework and leverage constrained spanning forests and path-packings to tighten guarantees, complemented by extensive experiments on 45 CLR benchmarks showing practical improvements over the prior $4.38$-approximation and proximity to optimal solutions. Collectively, the results advance both theoretical bounds and empirical performance for CLR and point to future refinements for related location-routing variants.

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.

Enhanced Approximation Algorithms for the Capacitated Location Routing Problem

TL;DR

This work advances the Capacitated Location Routing Problem (CLR) by introducing two approximation schemes. Tree-Alg achieves a -approximation for both unsplittable and splittable CLR, while Path-Alg attains a refined -approximation for splittable CLR, each supported by novel lower bounds and reductions to UFL. The authors develop an α-parameterized framework and leverage constrained spanning forests and path-packings to tighten guarantees, complemented by extensive experiments on 45 CLR benchmarks showing practical improvements over the prior -approximation and proximity to optimal solutions. Collectively, the results advance both theoretical bounds and empirical performance for CLR and point to future refinements for related location-routing variants.

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 -approximation algorithm for this problem, improving the best-known -approximation ratio. Moreover, if the demand of each customer is allowed to be delivered by multiple tours, we propose a more refined -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.

Paper Structure

This paper contains 20 sections, 14 theorems, 10 equations, 3 figures, 9 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Results
  4. Paper Organization
  5. Preliminaries
  6. Formal Problem Definition
  7. Lower Bounds
  8. General CLR
  9. The Algorithm
  10. The Analysis
  11. Splittable CLR
  12. The Algorithm
  13. The Analysis
  14. Experimental Results
  15. Instances
  16. ...and 5 more sections

Key Result

Lemma 3

It holds that $\hbox{OPT}'+(1-\alpha)\cdot\phi^*\leq \hbox{OPT}$.

Figures (3)

  • Figure 1: An illustration of a constrained spanning forest, constrained spanning path-packing, and constrained spanning cycle-packing, where each cycle node represents a customer and each square node represents a depot.
  • Figure 2: An illustration of the constrained spanning path-packing algorithm in Lemma \ref{['pathpack']}, where each cycle node represents a customer, each square node represents a depot, and the triangle node represents the super-depot.
  • Figure 3: An example of the path-splitting procedure, where each cycle node represents a customer and each square node represents a depot.

Theorems & Definitions (27)

  • Definition 1: CLR
  • Definition 2: UFL
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5: Lemma 3 in HarksKM13
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • ...and 17 more