Improved Approximation Algorithms for the Multiple-Depot Split Delivery Vehicle Routing Problem
The multiple-depot split delivery vehicle routing problem is a challenging optimization problem with broad applications in logistics and transportation. The goal is to serve customers' demand using a limited fleet of capacitated vehicles stationed at multiple depots, allowing each customer's demand to be split and served by multiple vehicles, while minimizing the total travel cost. Parameterized by the number of depots, the previous best-known result was a slice-wise polynomial-time $6$-approximation algorithm (INFORMS J. Comput. 2023), and whether this ratio could be improved remained an open question. We resolve this question by proposing a fixed-parameter tractable (FPT) $(2α+1+\varepsilon)$-approximation algorithm for any constant $\varepsilon>0$, where $α<3/2$ denotes the best-known ratio for the traveling salesman problem (TSP). Our algorithm enumerates partitions of connected components formed by low-cost edges to construct a low-cost cycle cover, then extracts paths, and assigns them to vehicles through a minimum-cost flow method. The cycle-cover technique also yields an FPT $(α+\varepsilon)$-approximation for the multiple-depot TSP. We further propose a simple parameterized $5$-approximation algorithm based on the structural properties of vehicle capacities, which achieves polynomial running time for a specific setting that appears in existing benchmark instances. In addition, we develop a bi-factor approximation algorithm that balances minor vehicle capacity violations against reductions in travel cost or gains in computational efficiency. Finally, our computational experiments demonstrate that the proposed methods exhibit complementary strengths across various instance types and achieve competitive solution quality.