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 $0<\varepsilon\le1/9$, applicable to general capacity and arbitrary positive demands. Our algorithm uses local search to select routes through one client with large demand and clients with small demands. It constructs candidate routes using endpoint quota trees after rounding demands and creating vertex copies. We combine these routes with those obtained by matching and set packing, keeping their client sets disjoint. The remaining clients are served by partitioning a tour constructed using weighted graphic matroid parity.

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