Improved Approximations for Capacitated Vehicle Routing with Unsplittable Client Demands

In this paper, we present improved approximation algorithms for the (unsplittable) Capacitated Vehicle Routing Problem (CVRP) in general metrics. In CVRP, introduced by Dantzig and Ramser (1959), 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 is 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 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 LP rounding with approximation guarantee $$\alpha +\ln (2) + \delta \approx 3.194 + \delta $$ in $$n^{O(1/\delta )}$$ time. Both approximations can further be improved by a small amount when combined with recent work by Blauth, Traub, and Vygen (2021), who obtained an $$(\alpha + 2\cdot (1 -\varepsilon ))$$ -approximation for unsplittable CVRP for some constant $$\varepsilon $$ depending on $$\alpha $$ ( $$\varepsilon > 1/3000$$ for $$\alpha = 1.5$$ ).