We study a trajectory analysis problem we call the Trajectory Capture Problem (TCP), in which, for a given input set ${\cal T}$ of trajectories in the plane, and an integer $k\geq 2$, we seek to compute a set of $k$ points (``portals'') to maximize the total weight of all subtrajectories of ${\cal T}$ between pairs of portals. This problem naturally arises in trajectory analysis and summarization. We show that the TCP is NP-hard (even in very special cases) and give some first approximation results. Our main focus is on attacking the TCP with practical algorithm-engineering approaches, including integer linear programming (to solve instances to provable optimality) and local search methods. We study the integrality gap arising from such approaches. We analyze our methods on different classes of data, including benchmark instances that we generate. Our goal is to understand the best performing heuristics, based on both solution time and solution quality. We demonstrate that we are able to compute provably optimal solutions for real-world instances.

中文翻译：

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探测一组轨迹以最大化捕获的信息

我们研究了一个轨迹分析问题，我们称之为轨迹捕获问题 (TCP)，其中，对于给定的平面中轨迹输入集 ${\cal T}$ 和整数 $k\geq 2$，我们寻求计算一组 $k$ 点（“门户”）以最大化门户对之间 ${\cal T}$ 的所有子轨迹的总权重。这个问题在轨迹分析和总结中自然会出现。我们证明 TCP 是 NP-hard 的（即使在非常特殊的情况下）并给出一些初步近似结果。我们的主要重点是使用实用的算法工程方法攻击 TCP，包括整数线性规划（解决实例以证明最优性）和局部搜索方法。我们研究了由这些方法引起的完整性差距。我们在不同类别的数据上分析我们的方法，包括我们生成的基准实例。我们的目标是根据求解时间和求解质量来了解性能最佳的启发式方法。我们证明我们能够为现实世界的实例计算可证明的最佳解决方案。