We study the problem of recovering the relative positions of objects moving along the real line based only on pairwise collision data. While interaction-based sensing systems arise naturally in a variety of practical settings, a systematic theoretical understanding of positional identifiability from collision observations alone remains unexplored. Our contributions are three-fold. First, under the full observability model, in which both the set of collisions and their temporal ordering are known, we show that the relative positions of all objects can be uniquely recovered if and only if the collision history, represented as a graph, is connected. Second, we show that under partial observability, where only colliding pairs are observed without timing information, the problem is related to \emph{function graphs} and introduce a canonical layer decomposition in which each layer corresponds to a maximal clique; the contraction graph induced by this decomposition is an interval graph, and we provide efficient algorithms to recover it. Third, under incomplete observations where even some pairwise collision observations may be missing, we formulate the problem as a graph completion problem and establish its NP-hardness via a $4$-approximation relationship with the graph bandwidth problem.

翻译：我们研究仅基于成对碰撞数据恢复沿实轴运动的物体相对位置的问题。尽管基于交互的传感系统在各种实际场景中自然出现，但仅从碰撞观测进行位置可辨识性的系统性理论理解仍未被探索。我们的贡献包含三方面。首先，在完全可观测模型下（即已知碰撞集合及其时间顺序），我们证明当且仅当以图表示的碰撞历史是连通图时，所有物体的相对位置能被唯一恢复。其次，我们证明在部分可观测条件下（仅观测到碰撞对而无时间信息），该问题与函数图相关，并引入一种规范层分解方法，其中每一层对应一个极大团；该分解诱导的收缩图是区间图，我们提供高效算法来恢复该图。第三，在不完全观测场景中（部分成对碰撞数据可能缺失），我们将该问题建模为图补全问题，并通过与图带宽问题的4-近似关系证明其NP难度。