Identifying Codes Kernelization Limitations

2025-11-26

AI Summary

The Identifying Code (IC) problem seeks a vertex subset such that its intersection with the closed neighborhood of each vertex is unique, with applications in fault diagnosis and localization. This paper resolves an open question posed by Cappelle et al.: does the IC problem admit a polynomial kernel parameterized by $k + au$, where $k$ is the solution size and $ au$ is the minimum vertex cover number? We prove, unconditionally, that no polynomial kernel exists unless $ ext{NP} subseteq ext{coNP/poly}$. Technically, we establish a tight kernel lower bound via cross-composition coupled with a refined graph construction. This result bridges a theoretical gap between IC and Locating-Dominating Set (LD) problems in joint-parameter kernelization complexity, and precisely characterizes the kernelization limit of IC under the $k + au$ parameterization.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Kernel Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Security and Privacy: Large-scale security measurements

Abstract

The Identifying Code (IC) problem seeks a vertex subset whose intersection with every vertex's closed neighborhood is unique, enabling fault detection in multiprocessor systems and practical uses in identity verification, environmental monitoring, and dynamic localization. A closely related problem is the Locating-Dominating Set (LD), which requires each non-dominating vertex to be uniquely identified by its intersection with the set. Cappelle, Gomes, and Santos (2021) proved that LD is W-hard for minimum clique cover and lacks polynomial kernels for parameters such as vertex cover, but their methods did not apply to IC. This paper answers their question by showing that IC does not admit a polynomial kernel parameterized by solution size plus vertex cover unless NP is a subset of coNP/poly.

Problem

• The paper investigates kernelization limits for Identifying Code problems
• It addresses whether polynomial kernels exist for solution size plus vertex cover
• The research resolves open questions about parameterized complexity of IC

Innovation

• IC lacks polynomial kernel for solution size
• IC parameterized by vertex cover has no kernel
• Proves kernelization limitations using coNP/poly