[Paper Review] Polynomial-Time Approximation Schemes for Bounded-Capacity Vehicle Routing and Clustering Problems in Metrics with Bounded Highway Dimension

Amariah Becker, Philip N. Klein+1 more|arXiv (Cornell University)|Jul 26, 2017

Facility Location and Emergency Management32 references3 citations

TL;DR

This paper presents polynomial-time approximation schemes (PTAS) for bounded-capacity vehicle routing and clustering problems in metrics with bounded highway dimension. By embedding such metrics into bounded-treewidth graphs with small additive error, the authors enable efficient approximation algorithms for single- and multiple-depot vehicle routing, k-center, and k-median problems with fixed k.

ABSTRACT

The concept of bounded highway dimension was developed to capture observed properties of the metrics of road networks. We show that a metric with bounded highway dimension and a distinguished point (the depot) can be embedded into a a graph of bounded treewidth in such a way that the distance between $u$ and $v$ is preserved up to an additive error of $\epsilon$ times the distance from $u$ or $v$ to the depot. We show that this theorem yields a PTAS for Bounded-Capacity Vehicle Routing in metrics of bounded highway dimension. In this problem, the input specifies a depot and a set of client locations; the output is a set of depot-to-depot tours, where each tour is responsible for visiting at most $Q$ client locations. Our PTAS can be extended to handle penalties for unvisited clients. We extend this embedding in a case where there is a set $S$ of distinguished points. The treewidth depends on $|S|$, and the distance between $u$ and $v$ is preserved up to an additive error of $\epsilon$ times the distance from $u$ and $v$ to $S$. This embedding implies a PTAS for Multiple Depot Bounded-Capacity Vehicle Routing, where the tours can go from one depot to an other. The embedding also implies that, for fixed $k$, there is a PTAS for $k$-Center in metrics of bounded highway dimension. In this problem, the goal is to minimize $d$ such that there exist $k$ points (the centers) such that every point is within distance $d$ of some center. Similarly, for fixed $k$, there is a PTAS for $k$-Median in metrics of bounded highway dimension. The goal of this problem is to minimize the sum of distances to the $k$ centers.

Motivation & Objective

• To develop efficient approximation algorithms for vehicle routing and clustering problems in road network-like metrics.
• To address bounded-capacity vehicle routing with a single or multiple depots.
• To extend the framework to k-center and k-median clustering with fixed k.
• To leverage the bounded highway dimension property to design scalable approximation schemes.
• To provide theoretical guarantees through a novel embedding into low-treewidth graphs with bounded distortion.

Proposed method

• Embed metrics with bounded highway dimension into graphs of bounded treewidth, preserving distances up to an additive error of ε times the distance to the depot or a set S of distinguished points.
• Use the treewidth structure to design dynamic programming-based approximation schemes for routing and clustering problems.
• Apply the embedding to single-depot and multiple-depot vehicle routing, allowing tours to start and end at any depot.
• Extend the approach to k-center and k-median problems by leveraging the low-treewidth structure for efficient optimization.
• Introduce a parameterized additive error that depends on the distance to the nearest depot or set S, ensuring controlled approximation quality.
• Use the bounded highway dimension to control the treewidth of the embedded graph, enabling polynomial-time algorithms.

Experimental results

Research questions

• RQ1 Can bounded-capacity vehicle routing in metrics with bounded highway dimension be approximated efficiently with a PTAS?
• RQ2 Does embedding bounded highway dimension metrics into low-treewidth graphs preserve distances with controllable error?
• RQ3 Can the embedding technique be extended to handle multiple depots in vehicle routing problems?
• RQ4 Is there a PTAS for k-center and k-median clustering in bounded highway dimension metrics when k is fixed?
• RQ5 What is the trade-off between approximation quality and the treewidth of the embedded graph?

Key findings

• A PTAS exists for bounded-capacity vehicle routing in metrics with bounded highway dimension, using a treewidth-based embedding with additive error.
• The embedding preserves distances up to an additive error of ε times the distance to the depot, enabling efficient dynamic programming on the low-treewidth graph.
• The framework extends to multiple-depot vehicle routing, allowing tours between depots while maintaining the PTAS guarantee.
• For fixed k, a PTAS is achieved for the k-center problem in bounded highway dimension metrics.
• For fixed k, a PTAS is also achieved for the k-median problem in such metrics, using the same embedding technique.
• The treewidth of the embedded graph depends on the number of distinguished points S, and the error term scales with the distance to S.