An Extension of the Das and Mathieu QPTAS to the Case of Polylog Capacity Constrained CVRP in Metric Spaces of a Fixed Doubling Dimension

Abstract

The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem having numerous practically important applications. CVRP is strongly NP-hard (even on the Euclidean plane), hard to approximate in the general case and APX-complete for an arbitrary metric. Meanwhile, for the geometric settings of the problem, there are known a number of quasi-polynomial and even polynomial time approximation schemes. Among these results, the well-known QPTAS proposed by A. Das and C. Mathieu appears to be the most general. In this paper, we propose the first extension of this scheme to a more wide class of metric spaces. Actually, we show that the metric CVRP has a QPTAS any time when the problem is set up in the metric space of any fixed doubling dimension \(d>1\) and the capacity does not exceed \(\mathrm {polylog}\,{(}n)\).