We introduce a new bounding approach called Continuity* C*, which provides optimality guarantees for the Moving-Target Traveling Salesman Problem (MT-TSP). Our approach relaxes the continuity constraints on the agent's tour by partitioning the targets' trajectories into smaller segments. This allows the agent to arrive at any point within a segment and depart from any point in the same segment when visiting each target. This formulation enables us to pose the bounding problem as a Generalized Traveling Salesman Problem (GTSP) on a graph, where the cost of traveling along an edge requires solving a new problem called the Shortest Feasible Travel (SFT). We present various methods for computing bounds for the SFT problem, leading to several variants of C*. We first prove that the proposed algorithms provide valid lower-bounds for the MT-TSP. Additionally, we provide computational results to validate the performance of all C* variants on instances with up to 15 targets. For the special case where targets move along straight lines, we compare our C* variants with a mixed-integer Second Order Conic Program (SOCP) based method, the current state-of-the-art solver for the MT-TSP. While the SOCP-based method performs well on instances with 5 and 10 targets, C* outperforms it on instances with 15 targets. For the general case, on average, our approaches find feasible solutions within approximately 4.5% of the lower-bounds for the tested instances.

翻译：本文提出了一种名为连续性*（C*）的新边界求解方法，该方法为移动目标旅行商问题（MT-TSP）提供了最优性保证。我们的方法通过将目标轨迹分割为更小的区段，从而放宽了智能体路径的连续性约束。这使得智能体在访问每个目标时，可以抵达轨迹区段内的任意点，并从同一区段内的任意点离开。该建模方式使我们能够将边界问题转化为图上的广义旅行商问题（GTSP），其中沿边移动的成本需要通过求解一个称为最短可行路径（SFT）的新问题来计算。我们提出了多种计算SFT问题边界的方法，从而衍生出C*的若干变体。我们首先证明了所提算法能为MT-TSP提供有效的下界。此外，我们通过计算结果验证了所有C*变体在最多包含15个目标的算例上的性能。针对目标沿直线移动的特殊情况，我们将C*变体与基于混合整数二阶锥规划（SOCP）的方法进行了比较，后者是当前MT-TSP领域最先进的求解器。虽然基于SOCP的方法在包含5个和10个目标的算例上表现良好，但C*在包含15个目标的算例上表现更优。对于一般情况，平均而言，我们的方法在测试算例中获得的可行解与下界的差距约为4.5%。