Improved Approximations for the Unsplittable Capacitated Vehicle Routing Problem

The capacitated vehicle routing problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. In this problem, we are given a depot and a set of customers, each with a demand, embedded in a metric space. The objective is to find a set of tours, each starting and ending at the depot, operated by the capacititated vehicle at the depot to serve all customers, such that all customers are served, and the total travel cost is minimized. We consider the unplittable variant, where the demand of each customer must be served entirely by a single tour. Let $α$ denote the current best-known approximation ratio for the metric traveling salesman problem. The previous best approximation ratio was $α+1+\ln 2+δ<3.1932$ for a small constant $δ>0$ (Friggstad et al., Math. Oper. Res. 2025), which can be further improved by a small constant using the result of Blauth, Traub, and Vygen (Math. Program. 2023). In this paper, we propose two improved approximation algorithms. The first algorithm focuses on the case of fixed vehicle capacity and achieves an approximation ratio of $α+1+\ln\bigl(2-\frac{1}{2}y_0\bigr)<3.0897$, where $y_0>0.39312$ is the unique root of $\ln\bigl(2-\frac{1}{2}y\bigr)=\frac{3}{2}y$. The second algorithm considers general vehicle capacity and achieves an approximation ratio of $α+1+y_1+\ln\left(2-2y_1\right)+δ<3.1759$ for a small constant $δ>0$, where $y_1>0.17458$ is the unique root of $\frac{1}{2} y_1+ 6 (1-y_1)\bigl(1-e^{-\frac{1}{2} y_1}\bigr) =\ln\left(2-2y_1\right)$. Both approximations can be further improved by a small constant using the result of Blauth, Traub, and Vygen (Math. Program. 2023).

💡 Research Summary

The paper addresses the unsplittable Capacitated Vehicle Routing Problem (CVRP), where each customer’s demand must be served entirely by a single vehicle tour. The authors improve upon the previous best approximation ratios, which stood at α + 1 + ln 2 ≈ 3.1932 (Friggstad et al., 2025) and could be marginally reduced using the technique of Blauth, Traub, and Vygen (2023). Here α denotes the best known approximation factor for the metric Traveling Salesman Problem (TSP).

Two new algorithms are presented. The first targets the case of a fixed vehicle capacity k. It runs in O(n·k) time and achieves an approximation ratio of

 α + 1 + ln(2 − ½ y₀) < 3.0897,

where y₀ ≈ 0.39312 is the unique root of the equation ln(2 − ½ y) = (3/2) y. The second algorithm works for arbitrary capacities and runs in O(n^{1/δ}) time (for any small δ > 0). Its ratio is

 α + 1 + y₁ + ln(2 − 2 y₁) + δ < 3.1759,

with y₁ ≈ 0.17458 being the unique solution of ½ y + 6(1 − y)(1 − e^{−y/2}) = ln(2 − 2 y). Both ratios can be further lowered by a constant using the result of Blauth et al. (2023).

The technical contribution rests on a refined version of the δ‑ITP algorithm, called δ‑ITP+. The original δ‑ITP (Friggstad et al., 2025) partitions a TSP tour into paths and serves small‑demand customers (d ≤ δ) differently from larger ones. δ‑ITP+ first serves all “large” customers (d > ½ k) with trivial tours (depot‑customer‑depot) and then applies δ‑ITP to the remaining small and medium customers. This simple preprocessing strictly improves the cost bound of Lemma 3 in the original work.

For the fixed‑capacity algorithm, the authors combine δ‑ITP+ with two sub‑algorithms. SubAlg.1 follows the combinatorial approach of Friggstad et al.: it matches customers with demand > k/3, serves them, and then uses 1/3‑ITP on the rest. SubAlg.2 formulates a linear program whose variables correspond to all O(n·k) feasible tours (each serving at most k customers). After solving the LP, a randomized rounding selects a subset of tours; the uncovered customers are finally handled by 1/3‑ITP+. By analyzing the cost contributions of both sub‑algorithms, the authors prove that the better of the two yields the claimed ratio α + 1 + ln(2 − ½ y₀).

When the vehicle capacity is not fixed, the number of possible tours can be exponential, so the authors restrict attention to O(n^{1/δ}) tours that serve customers with demand larger than δ·k. SubAlg.3 solves an LP over these tours, rounds it, and finishes with 1/3‑ITP+. SubAlg.4 is similar but uses the general δ‑ITP+ for the remaining customers. Together with SubAlg.1, the algorithm selects the best solution among the three, guaranteeing the ratio α + 1 + y₁ + ln(2 − 2 y₁) + δ.

The analysis hinges on careful lower bounds for the optimal CVRP cost (Lemma 2) and precise upper bounds for each algorithmic component. The constants y₀ and y₁ arise from solving equations that balance the contribution of the LP‑rounded tours against the cost of the ITP‑style partitioning. Numerical evaluation shows y₀ ≈ 0.39312 and y₁ ≈ 0.17458, leading to the concrete ratios 3.0897 and 3.1759 when α = 1.5 (the Christofides bound). Moreover, by invoking the improvement technique of Blauth et al., both ratios can be reduced by an additional small constant.

In summary, the paper delivers two practically implementable algorithms that improve the state‑of‑the‑art approximation factors for the unsplittable CVRP by roughly 0.1–0.2. The work advances the theoretical understanding of CVRP approximability, introduces a useful refinement of the ITP framework, and opens avenues for further improvements via tighter LP formulations or specialized metric spaces.