QPTAS for the CVRP with a Moderate Number of Routes in a Metric Space of Any Fixed Doubling Dimension

Abstract

The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem having a host of valuable practical applications in operations research. The CVRP is strongly NP-hard both in its general case and even in very specific settings (e.g., on the Euclidean plane). The problem is APX-complete for an arbitrary metric and admits Quasi-Polynomial Time Approximation Scheme (QPTAS) in the Euclidean space of any fixed dimension (and even PTAS, under additional constraints). In this paper, we significantly extend the class of metric settings of the CVRP that can be approximated efficiently. We show that the metric CVRP admits QPTAS any time, when it is formulated in a metric space of a fixed doubling dimension \(d>1\) and is restricted to have an optimal solution of at most \(\mathrm {polylog}\,{n}\) routes.