A Framework for Vehicle Routing Approximation Schemes in Trees

Abstract

We develop a general framework for designing polynomial-time approximation schemes (PTASs) for various vehicle routing problems in trees. In these problems, the goal is to optimally route a fleet of vehicles, originating at a depot, to serve a set of clients, subject to various constraints. For example, in Minimum Makespan Vehicle Routing, the number of vehicles is fixed, and the objective is to minimize the longest distance traveled by a single vehicle. Our main insight is that we can often greatly restrict the set of potential solutions without adding too much to the optimal solution cost. This simplification relies on partitioning the tree into clusters such that there exists a near-optimal solution in which every vehicle that visits a given cluster takes on one of a few forms. In particular, only a small number of vehicles serve clients in any given cluster. By using these coarser building blocks, a dynamic programming algorithm can find a nearoptimal solution in polynomial time. We show that the framework is flexible enough to give PTASs for many problems, including Minimum Makespan Vehicle Routing, Distance-Constrained Vehicle Routing, Capacitated Vehicle Routing, and School Bus Routing, and can be extended to the multiple depot setting.

A PTAS for Capacitated Vehicle Routing on Trees

“…In Section 2, we decompose the tree into components and we show that there exists a nearoptimal solution that is simple inside the components. The decomposition uses the novel technique of tree condensing introduced by Becker and Paul [BP19], but with a different choice of parameters. Becker and Paul [BP19] use small clusters, such that terminals within a leaf cluster are covered by just one tour, whereas we use components, such that terminals within a leaf component are covered by a constant O (1) number of tours.…”

“…More recently, researchers tried to go beyond a constant factor so as to get a (1 + )-approximation, at the cost of relaxing some of the constraints. When the capacity of the tour is allowed to be violated by an fraction, there is a bicriteria PTAS for the unit demand tree CVRP due to Becker and Paul [BP19]. When the running time is allowed to be quasi-polynomial, Jayaprakash and Salavatipour [JS22] very recently gave a quasi-polynomial time approximation scheme (QPTAS) for the unit demand and the splittable tree CVRP.…”

Improved Approximations for CVRP with Unsplittable Demands

“…Later, Becker [Bec18] gave a 4/3-approximation with respect to the lower bound. Becker and Paul [BP19] showed a (1, 1 + ε)-bicriteria polynomialtime approximation scheme for splittable CVRP in trees, i.e. a PTAS but every tour serves at most (1 + ε)Q demand.…”

“…Recently, Jayaprakash and Salavatirpour [JS] presented a QPTAS for unitdemand CVRP for trees and more generally graphs of bounded treewidth, bounded doubling metrics, or bounded highway dimension. Even more recently, building upon ideas of [BP19] and [JS], Mathieu and Zhou [MZ20] have presented a PTAS for splittable CVRP on trees.…”

A PTAS for the Min-Max Euclidean Multiple TSP

“…Becker and Paul considered multiple TSP in the context of navigation in trees [BP19]. Their objective is the same as ours, but distances are measured as path lengths in a tree.…”