Improved Approximation for Unsplittable Capacitated Vehicle Routing Using Quota Trees

Overview & Synopsis

In the Unsplittable Capacitated Vehicle Routing Problem (UCVRP), we are given a set $V$ of $n$ clients with normalized demands $d_v \in (0,1]$, a depot $r$, a unit vehicle capacity, and symmetric metric travel costs $c$ on $V \cup \{r\}$. We seek a collection of routes of minimum total cost that start and end at $r$, serve each client's entire demand on a single route, and respect the unit capacity constraint $\sum_{v \in V(R)} d_v \le 1$ on every route $R$, where $V(R)$ denotes the clients served by $R$. This models deliveries from a common depot in which each customer's order must be delivered in a single trip. UCVRP generalizes the metric traveling salesman problem (TSP) and is therefore NP-hard. Capacity constraints add a packing difficulty: when all clients are colocated away from the depot, minimizing the route cost is equivalent to minimizing the number of bins in bin packing. Consequently, UCVRP admits no polynomial-time approximation ratio below $3/2$ unless $\textbf{P=NP}$ (Grandoni, Mathieu, and Zhou, 2023). The classical $3.5$-approximation has been improved through several recent advances; in particular, Zhao and Xiao (2026) recently obtained a $(2+\ln 2+\varepsilon)$-approximation. In this paper, we give a deterministic $(5/2+\varepsilon)$-approximation algorithm running in $n^{O(1/\varepsilon)}$ time for every rational $0