We introduce a new bounding approach called Continuity* (C*) that provides optimality guarantees to the Moving-Target Traveling Salesman Problem (MT-TSP). Our approach relies on relaxing the continuity constraints on the agent's tour. This is done by partitioning the targets' trajectories into small sub-segments and allowing the agent to arrive at any point in one of the sub-segments and depart from any point in the same sub-segment when visiting each target. This lets us pose the bounding problem as a Generalized Traveling Salesman Problem (GTSP) in a graph where the cost of traveling an edge requires us to solve a new problem called the Shortest Feasible Travel (SFT). We also introduce C*-lite, which follows the same approach as C*, but uses simple and easy to compute lower-bounds to the SFT. We first prove that the proposed algorithms provide lower bounds to the MT-TSP. We also provide computational results to corroborate the performance of C* and C*-lite for instances with up to 15 targets. For the special case where targets travel along lines, we compare our C* variants with the SOCP based method, which is the current state-of-the-art solver for MT-TSP. While the SOCP based method performs well for instances with 5 and 10 targets, C* outperforms the SOCP based method for instances with 15 targets. For the general case, on average, our approaches find feasible solutions within ~4% of the lower bounds for the tested instances.

翻译：我们提出了一种名为Continuity*（C*）的新型边界方法，可为移动目标旅行商问题（MT-TSP）提供最优性保证。该方法通过松弛智能体路径的连续性约束来实现：将目标的轨迹划分为小子段，并允许智能体在访问每个目标时，从同一子段内的任意点抵达并从中任意点离开。由此我们将边界问题转化为广义旅行商问题（GTSP），其中边的旅行成本需要求解一个称为最短可行行程（SFT）的新问题。我们还提出了C*-lite方法，它沿用C*的框架，但采用简单易计算的SFT下界。我们首先证明所提算法为MT-TSP提供下界，并通过计算实验验证C*和C*-lite在最多15个目标实例上的性能。针对目标沿直线运动的特例，我们将其与当前MT-TSP最先进的求解器（基于SOCP的方法）进行比较。对于5个和10个目标的实例，SOCP方法表现良好；但在15个目标实例中，C*的性能超越SOCP方法。对于一般情况，我们的方法在测试实例中平均能找到距离下界约4%以内的可行解。