Improved Approximations for a CVRP with Unsplittable Demands

In this paper, we present improved approximation algorithms for the (unsplittable) capacitated vehicle routing problem (CVRP) in general metrics. In the CVRP, we are given a set of points (clients) V together with a depot r in a metric space, with each $v\in V$ having a demand $d_v>0$ and a vehicle of bounded capacity Q. The goal is to find a minimum cost collection of tours for the vehicle, each starting and ending at the depot, such that each client is visited at least once and the total demands of the clients in each tour are at most Q. In the unsplittable variant we study, the demand of a node must be served entirely by one tour. We present two approximation algorithms for the unsplittable CVRP: a combinatorial $(\alpha +1.75)$-approximation, where $\alpha$ is the approximation factor for the traveling salesman problem, and an approximation algorithm based on linear programming rounding with approximation guarantee $\alpha +\ln (2)+\delta \approx 3.194+\delta$ in $n^{O(1/\delta )}$ time.

Funding: This work was supported by the Canadian Network for Research and Innovation in Machining Technology and the Natural Sciences and Engineering Research Council of Canada. Z. Friggstad was supported by an NSERC Discovery Grant and a Discovery Grant Accelerator. M. R. Salavatipoura was supported by an NSERC Discovery Grant.