Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity

Jingyang Zhao, Mingyu Xiao

Abstract: The Capacitated Vehicle Routing Problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. According to the property of the demand of customers, we distinguish three variants of CVRP: unit-demand, splittable and unsplittable. We consider $k$-CVRP in general metrics and general graphs, where $k$ is the capacity of the vehicle and all the three versions are APX-hard for each fixed $k\geq 3$. In this paper, we give a $(5/2-\Theta(\sqrt{1/k}))$-approximation algorithm for splittable and unit-demand $k$-CVRP and a $(5/2+\ln2-\Theta(\sqrt{1/k}))$-approximation algorithm for unsplittable $k$-CVRP (assume the approximation ratio for metric TSP is $\alpha=3/2$). Thus, our approximation ratio is better than previous results for sufficient large $k$, say $k\leq 1.7\times 10^7$. For small $k$, we design independent algorithms by using more techniques to get further improvements. For splittable and unit-demand cases, we improve the ratio from $1.934$ to $1.500$ for $k=3$, and from $1.750$ to $1.667$ for $k=4$. For the unsplittable case, we improve the ratio from $2.693$ to $1.500$ for $k=3$, from $2.443$ to $1.750$ for $k=4$, and from $2.893$ to $2.157$ for $k=5$.