Positional Identifiability from Pairwise Collision Data

2026-05-21

AI Summary

This study addresses the problem of reconstructing the relative positions of objects on the real line using only pairwise collision data. The authors develop a graph-theoretic framework and tailored recovery algorithms for three observation regimes: complete, partial, and incomplete. Under complete observability, they establish an equivalence between positional identifiability and the connectivity of the collision graph, enabling unique reconstruction. For partially observed data, they propose an efficient layer decomposition algorithm based on functional graphs. In the incompletely observed setting, they prove the problem is NP-hard and demonstrate a 4-approximation reduction to the graph bandwidth problem. Combining tools from graph theory, clique decomposition, and interval graph construction, the work systematically characterizes the solvability limits and provides a unified algorithmic framework for this inverse problem.

Abstract

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.

Problem

• positional identifiability
• pairwise collision data
• relative positions
• collision history
• graph observability

Innovation

• positional identifiability
• pairwise collision data
• function graphs
• interval graph
• graph completion