# A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs

## TL;DR

This paper introduces a polynomial-time approximation scheme for the Capacitated Vehicle Routing problem on planar graphs with bounded capacity, improving upon the previous quasipolynomial-time solutions by embedding graphs into bounded-treewidth structures.

## Contribution

It provides the first PTAS for this problem on planar graphs with capacity bounds, using a novel embedding technique to approximate client distances.

## Key findings

- Achieved a PTAS for bounded-capacity vehicle routing in planar graphs.
- Embedded planar graphs into bounded-treewidth graphs with minimal distance distortion.
- Improved computational complexity over previous quasipolynomial-time algorithms.

## Abstract

The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the client-to-client distances up to a small additive error proportional to client distances to the depot.