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Identifying Codes Kernelization Limitations

Aritra Banik, Praneet Kumar Patra, Adele Anna Rescigno, Abhishek Sahu

TL;DR

The paper resolves the open question of whether Identifying Code ($IC$) admits a polynomial kernel when parameterized by the combined size of the solution and the vertex cover. It employs an OR-cross-composition from Set Splitting into a single IC instance and invokes Bodlaender et al.'s kernelization framework to establish that a polynomial kernel does not exist unless $NP ⊆ coNP/poly$. The reduction constructs a graph $G$ with a vertex cover of size $O(|U|+log m+log t)$ and an Identifying Code of size at most $6n+r+11s+14$ iff some Set Splitting instance is a yes-instance. This result delineates a kernelization gap between IC and LD and has implications for preprocessing in localization and fault-detection applications.

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.

Identifying Codes Kernelization Limitations

TL;DR

The paper resolves the open question of whether Identifying Code () admits a polynomial kernel when parameterized by the combined size of the solution and the vertex cover. It employs an OR-cross-composition from Set Splitting into a single IC instance and invokes Bodlaender et al.'s kernelization framework to establish that a polynomial kernel does not exist unless . The reduction constructs a graph with a vertex cover of size and an Identifying Code of size at most iff some Set Splitting instance is a yes-instance. This result delineates a kernelization gap between IC and LD and has implications for preprocessing in localization and fault-detection applications.

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.

Paper Structure

This paper contains 2 sections, 4 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. No polynomial kernel for IC

Key Result

theorem 1

BodlaenderJK14 If an NP-hard problem $P$ OR-cross-composes into the parameterized problem $Q$, then $Q$ does not admit a polynomial kernelization unless $NP \subseteq coNP/poly$.

Figures (2)

  • Figure 1: Gadgets used in the construction of graph $G$
  • Figure 2: The construction of graph $G$ step by step. To avoid overloading, some edges within the gadgets have been removed. (a) The connection between the vertices in ${\bar{{\cal F}}}_+,{\bar{{\cal F}}}_-,{\bar{{\cal F}}}_*$ and in ${U}^-,{U}^+$. (b) The connection between the vertices in ${\bar{{\cal F}}}_+,{\bar{{\cal F}}}_-,{\bar{{\cal F}}}_*$ and in $\Delta_0,\Delta_1$

Theorems & Definitions (6)

  • theorem 1
  • theorem 2
  • lemma 1
  • proof
  • theorem 3
  • proof