Identifying Codes Kernelization Limitations

TL;DR

This paper proves that the Identifying Code problem does not have a polynomial kernel when parameterized by solution size plus vertex cover, unless a major complexity class collapse occurs.

Contribution

It extends kernelization lower bounds to the Identifying Code problem, which was previously unresolved.

Findings

IC does not admit a polynomial kernel under the given parameterization.

The result relies on complexity-theoretic assumptions (NP not in coNP/poly).

Addresses an open question from prior research.

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.