The Moving Target Vehicle Routing Problem with Obstacles (MT-VRP-O) seeks trajectories for several agents that collectively intercept a set of moving targets. Each target has one or more time windows where it must be visited, and the agents must avoid static obstacles and satisfy speed and capacity constraints. We introduce Lazy Branch-and-Price with Relaxed Continuity (Lazy BPRC), which finds optimal solutions for the MT-VRP-O. Lazy BPRC applies the branch-and-price framework for VRPs, which alternates between a restricted master problem (RMP) and a pricing problem. The RMP aims to select a sequence of target-time window pairings (called a tour) for each agent to follow, from a limited subset of tours. The pricing problem adds tours to the limited subset. Conventionally, solving the RMP requires computing the cost for an agent to follow each tour in the limited subset. Computing these costs in the MT-VRP-O is computationally intensive, since it requires collision-free motion planning between moving targets. Lazy BPRC defers cost computations by solving the RMP using lower bounds on the costs of each tour, computed via motion planning with relaxed continuity constraints. We lazily evaluate the true costs of tours as-needed. We compute a tour's cost by searching for a shortest path on a Graph of Convex Sets (GCS), and we accelerate this search using our continuity relaxation method. We demonstrate that Lazy BPRC runs up to an order of magnitude faster than two ablations.

翻译：移动目标车辆路径问题（MT-VRP-O）旨在为多个智能体寻找轨迹，使其共同拦截一组移动目标。每个目标需在其一个或多个时间窗口内被访问，且智能体需避开静态障碍物，同时满足速度与容量约束。我们提出具有松弛连续性的懒惰分支定价法（Lazy BPRC），该方法可为MT-VRP-O求得最优解。Lazy BPRC采用VRP的分支定价框架，在主问题受限版（RMP）与定价问题之间交替迭代。受限主问题从有限子集中为每个智能体选择一组目标-时间窗口配对序列（称为路径）。定价问题则向该有限子集添加新路径。传统方法求解RMP需计算每个路径的代价，而在MT-VRP-O中，该代价因需在移动目标间进行无碰撞运动规划而计算成本极高。Lazy BPRC通过基于连续松弛约束的运动规划计算各路径代价的下界来推迟真实代价计算，从而求解RMP。我们按需惰性评估路径的真实代价，通过在凸集图（GCS）上搜索最短路径计算路径代价，并利用连续性松弛方法加速搜索。实验表明，Lazy BPRC的运行速度比两种消融方法快一个数量级。