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：带障碍的运动目标车辆路径问题（MT-VRP-O）旨在为多个智能体规划轨迹，使其能够集体拦截一组运动目标。每个目标具有一个或多个必须被访问的时间窗口，且智能体需规避静态障碍物并满足速度与容量约束。我们提出采用松弛连续性的惰性分支定价算法（Lazy BPRC），可求解MT-VRP-O的最优解。Lazy BPRC应用了针对VRP的分支定价框架，该框架在受限主问题（RMP）与定价问题之间交替迭代。RMP旨在为每个智能体从有限子集中选择一组目标-时间窗口配对序列（称为旅行路线）。定价问题则负责向有限子集中添加新路线。传统方法求解RMP时需计算每个智能体沿有限子集中每条路线行驶的成本，而在MT-VRP-O中，由于需要规划运动目标间的无碰撞运动，这一成本计算计算量极大。Lazy BPRC通过使用基于松弛连续性约束的运动规划计算的各条路线成本下界来求解RMP，从而推迟成本计算。我们按需惰性评估路线的真实成本，通过搜索凸集图（GCS）上的最短路径计算路线成本，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变体快一个数量级。