Multidepot Capacitated Vehicle Routing with Improved Approximation Guarantees
Jingyang Zhao, Mingyu Xiao
TL;DR
This work advances approximation algorithms for the Multidepot Capacitated Vehicle Routing Problem (k-MCVRP) in metric graphs by delivering new guarantees for all demand variants (unsplittable, splittable, unit-demand) and both general and fixed-capacity regimes. The authors develop and blend three core techniques—cycle-partition, tree-partition, and LP-based rounding—to achieve tight improvements: a $4-1/1500$-approximation for splittable/unit-demand and $4-1/50000$ for unsplittable in general, and refined bounds of $3+\ln 2-\Theta(1/\sqrt{k})$ (with further $3+\ln 2-1/9000$ improvements for splittable/unit-demand when $k$ is fixed). The improvements hinge on refined tree-partition analysis, LP-rounding strategies, and careful case analyses that balance cycle- and tree-based constructions. Collectively, these results push the boundary on fixed-destination k-MCVRP and offer practical, near-optimal strategies with clear implications for multi-depot logistics optimization.
Abstract
The Multidepot Capacitated Vehicle Routing Problem (MCVRP) is a well-known variant of the classic Capacitated Vehicle Routing Problem (CVRP), where we need to route capacitated vehicles located in multiple depots to serve customers' demand such that each vehicle must return to the depot it starts, and the total traveling distance is minimized. There are three variants of MCVRP according to the property of the demand: unit-demand, splittable and unsplittable. We study approximation algorithms for $k$-MCVRP in metric graphs, where $k$ is the capacity of each vehicle. The best-known approximation ratios for the three versions are $4-Θ(1/k)$, $4-Θ(1/k)$, and $4$, respectively. We give a $(4-1/1500)$-approximation algorithm for unit-demand and splittable $k$-MCVRP, and a $(4-1/50000)$-approximation algorithm for unsplittable $k$-MCVRP. When $k$ is a fixed integer, we give a $(3+\ln2-\max\{Θ(1/\sqrt{k}),1/9000\})$-approximation algorithm for the splittable and unit-demand cases, and a $(3+\ln2-Θ(1/\sqrt{k}))$-approximation algorithm for the unsplittable case.