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Approximating Multiple-Depot Capacitated Vehicle Routing via LP Rounding

Zachary Friggstad, Tobias Mömke

TL;DR

This work introduces a polynomial-time randomized 3.9365-approximation for Capacitated Vehicle Routing with Multiple Depots (CVRP-MD) via a novel LP relaxation plus preflow-based path sampling. The method constructs R-rooted paths that cover most clients with cost near a scaled opt_LP, then grafts the remaining clients through a cheap forest and a careful tree-pruning and path-splitting scheme to form feasible tours with at most k clients each. The analysis balances path costs, radial lower bound contributions, and forest costs using fixed constants and a delta parameter that captures how close lb is to opt, yielding the 3.9365 − 0.49826 δ bound. The approach is compatible with recent CVRP improvements and offers a concrete path to small constant improvements, contributing a significant advance in CVRP-MD approximations with a tractable LP-based framework.

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.

Approximating Multiple-Depot Capacitated Vehicle Routing via LP Rounding

TL;DR

This work introduces a polynomial-time randomized 3.9365-approximation for Capacitated Vehicle Routing with Multiple Depots (CVRP-MD) via a novel LP relaxation plus preflow-based path sampling. The method constructs R-rooted paths that cover most clients with cost near a scaled opt_LP, then grafts the remaining clients through a cheap forest and a careful tree-pruning and path-splitting scheme to form feasible tours with at most k clients each. The analysis balances path costs, radial lower bound contributions, and forest costs using fixed constants and a delta parameter that captures how close lb is to opt, yielding the 3.9365 − 0.49826 δ bound. The approach is compatible with recent CVRP improvements and offers a concrete path to small constant improvements, contributing a significant advance in CVRP-MD approximations with a tractable LP-based framework.

Abstract

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

Paper Structure

This paper contains 15 sections, 13 theorems, 23 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Review of Previous Approximations
  4. Our Approach
  5. CVRP-MD Algorithm
  6. Step: Preflow Rounding
  7. Step: Covering the Uncovered Nodes with Trees Rooted in $\mathcal{P}$
  8. Algorithm Summary
  9. Step: Tree Pruning
  10. Step: Forming the Final Tours
  11. Approximation Guarantee Analysis
  12. Compatibility with blauth2021improving and Further Improvements
  13. Completing the Analysis
  14. Proof of Lemma \ref{['lem:pruning_analysis']}
  15. Proof of Lemma \ref{['lem:splitting']}

Key Result

Theorem 1

There is a polynomial-time randomized algorithm for CVRP-MD that finds a solution whose expected cost is at most $3.9365$ times the optimum solution's cost.

Figures (2)

  • Figure 1: Left: The black nodes were already covered by a previous pruning step and the grey nodes are the ones to be covered in the tour $\mathcal{T}(G_i,G_j)$ (so $k = 10$). All nodes of $U(G)$ will be covered and the grey nodes in $G'$ are precisely $A$. The square node is the depot closest to a client in $U(G_i) \cup A$. Tour $\mathcal{T}(G_i,G_j)$ is obtained by doubling all edges shown in the picture and shortcutting to only include $U(G_i) \cup A$ and the depot. Right: All subtrees in $G_i$ are pruned, but all subtrees in $G_j$ remain. The nodes of $A$ are now covered.
  • Figure 2: Top: A path $P \in \mathcal{P}$, depicted horizontally, and the trees $T'$ of $F'$ with a root in $P$. The thick vertical edges are the trunks of the trees. The numbers indicate the final ordering of the nodes. The black nodes were covered earlier by tours in the tree pruning stage. With $k = 12$ and offset $\tau = 6$, the set consisting of the depot and $S_{P'}$, depicted with grey nodes, would be nodes $1, 6, 18$ and $30$ in this ordering. Bottom: The edges that were doubled to form the various tours. The white square nodes are depots nearest each node of $S_{P'}$. Notice each edge lies on at most one subtree with the following exceptions: for a large tree $T'$ some edges from a single path between $S_{P'}$ and the trunk appear on two subtrees and for a small tree $T'$ some edges on a single path from $S_{P'}$ to the root of the tree appear on two subtrees. A client that is not numbered in means we do not consider it as being covered by the corresponding tour.

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 1: Haimovich, Rinnooy Kan haimovich1985bounds
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 16 more