Optimal Error-detection system for Identifying Codes

TL;DR

This paper studies optimal error-detecting identifying codes in graphs, focusing on fault tolerance, and shows the NP-completeness of finding minimum such codes in general graphs.

Contribution

It introduces a fault-tolerant variant of identifying codes and analyzes their minimum size in specific graph classes, establishing NP-completeness for general graphs.

Findings

Minimum-sized error-detecting identifying codes are characterized for certain graph classes.

The problem is NP-complete in arbitrary graphs.

Fault-tolerant codes enhance detection reliability in network systems.

Abstract

Assume that a graph $G$ models a detection system for a facility with a possible ``intruder," or a multiprocessor network with a possible malfunctioning processor. We consider the problem of placing detectors at a subset of vertices in $G$ to determine the location of an intruder if there is any. Many types of detection systems have been defined for different sensor capabilities; in particular, we focus on Identifying Codes, where each detector can determine whether there is an intruder within its closed neighborhood. In this research we explore a fault-tolerant variant of identifying codes applicable to real-world systems. Specifically, error-detecting identifying codes permit a false negative transmission from any single detector. We investigate minimum-sized error-detecting identifying codes in several classes of graphs, including cubic graphs and infinite grids, and show that the problem of determining said minimum size in arbitrary graphs is NP-complete.