Identifying Codes and Covering Problems

• Published in: IEEE transactions on information theory Vol. 54; no. 9; pp. 3929 - 3950
• Main Authors: Laifenfeld, M., Trachtenberg, A.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Approximation; Approximation algorithms; Codes; Communication system control; Covering; Cryptography; Data encryption; Distributed algorithms; Environmental monitoring; Error correction codes; Exact sciences and technology; Fault detection; Fault diagnosis; Graphs; Hardness; hardness of approximation; identifying codes; Information processing; Information theory; Information, signal and communications theory; Joints; Mathematical analysis; Monitoring; robust identifying codes; Robustness; set cover; System testing; Telecommunications and information theory; test cover; Topology; Performance evaluation; Polynomial approximation; hardness of approximation; Two dimensional model; test cover; Polynomial time; identifying codes; robust identifying codes; set cover; Distributed algorithm; Defect localization; Environmental monitoring; Distributed algorithms; Fault diagnostic; Defect detection; Information theory
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2008.928263

The identifying code problem for a given graph involves finding a minimum set of vertices whose neighborhoods uniquely overlap at any given graph vertex. Initially introduced in 1998, this problem has demonstrated its fundamental nature through a wide variety of applications, such as fault diagnosis, location detection, and environmental monitoring, in addition to deep connections to information theory, superimposed and covering codes, and tilings. This work establishes efficient reductions between the identifying code problem and the well-known set-covering problem, resulting in a tight hardness of approximation result and novel, provably tight polynomial-time approximations. The main results are also extended to r -robust identifying codes and analogous set (2 r +1) -multicover problems. Finally, empirical support is provided for the effectiveness of the proposed approximations, including good constructions for well-known topologies such as infinite two-dimensional grids.