On a new class of codes for identifying vertices in graphs

Karpovsky, Mark G.; Chakrabarty, Krishnendu; Levitin, Lev B.

doi:10.1109/18.661507

Cited by 377 publications

(344 citation statements)

References 19 publications

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“…One of the interests of these notions lies in their applications to the location of threats in facilities [54,58] and error detection in computer networks [40]. One can also mention applications to routing [44], to bioinformatics [37] (where identifying codes are called differentiating dominating sets) and to measuring the first-order logical complexity of graphs [41] (where locating-dominating sets are calledsieves).…”

Section: Definitions and Problemsmentioning

confidence: 99%

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Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Foucaud

2015

Journal of Discrete Algorithms

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An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code. When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set. These notions are closely related to a number of similar and well-studied concepts such as the one of a test cover. In this paper, we study the decision problems Identifying Code and Locating-Dominating Set (which consist in deciding whether a given graph admits an identifying code or a locating-dominating set, respectively, with a given size) and their minimization variants Minimum Identifying Code and Minimum LocatingDominating Set. These problems are known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Moreover, it is known that they are approximable within a logarithmic factor, but hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite: both problems remain hard in these cases. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), the two problems are hard to approximate within some constant factor, a question which was open. We summarize all known results in the area, and we compare them to the ones for the related problem Dominating Set. In particular, our work exhibits important graph classes for which Dominating Set is efficiently solvable, but Identifying Code and Locating-Dominating Set are hard (whereas in all previous works, their complexity was the same). We also introduce graph classes for which the converse holds, and for which the complexities of Identifying Code and Locating-Dominating Set differ.

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Section: Definitions and Problemsmentioning

confidence: 99%

“…The notion that we will mainly study, introduced in 1998 in [40], is the one of an identifying code. Given a graph G and a vertex v of G, we denote by N (v) and by N [v] the open and the closed neighbourhood of v, respectively.…”

Section: Definitions and Problemsmentioning

confidence: 99%

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Foucaud

2015

Journal of Discrete Algorithms

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“…the identifying code problem is to find an identifying code of minimum cardinality for any given graph. Identifying codes have been studied extensively since their introduction in 1998 [5], and they have formed a fundamental basis for a wide variety of theoretical work and practical applications.…”

Section: A Identifying Codesmentioning

confidence: 99%

“…Applications such as fault detection and diagnosis in network and specifically in processor arrays [5], location detection in sensor networks [6,7], environmental monitoring in utility networks [10], and others, have been studied extensively, using different approaches that have resulted in a large variety of practical algorithms as well as theoretical results. In this paper we focus on a combinatorial approach, which is closely related to error-correcting codes, and specifically to identifying codes.…”

Section: Introductionmentioning

confidence: 99%

Localization and identification in networks using robust identifying codes

Laifenfeld

2008

2008 Information Theory and Applications Workshop

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Abstract-Many practical problems in networks such as fault detection and diagnosis, location detection, environmental monitoring, etc, require to identify specific nodes or links based on a set of observations, which size is a subject of optimization. In this paper we focus on a combinatorial approach to these problems, which is closely related to coding techniques, and specifically to identifying codes. The identifying code problem for a given graph involves finding a minimum set of vertices whose neighborhoods uniquely overlap at any given graph vertex. In this paper we show efficient reductions between the identifying code problem and well-known covering problems, resulting in a tight hardness of approximation result and provable good centralized and distributed approximations. We further provide empirical and theoretical results on identifying codes in random networks and efficient constructions of these codes in infinite grids.

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“…Such a subset C ⊆ V satisfying the two requirements is called an identifying code (any nonempty subset of V is called a code). The concept of identifying codes was introduced in [10], where a fault diagnosis was performed in multiprocessor networks.…”

Section: Introductionmentioning

confidence: 99%

Tolerant identification with Euclidean balls

2012

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The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. The identifying codes can be applied, for example, to sensor networks. In this paper, we consider as sensors the set Z 2 where one sensor can check its neighbours within Euclidean distance r. We construct tolerant identifying codes in this network that are robust against some changes in the neighbourhood monitored by each sensor. We give bounds for the smallest density of a tolerant identifying code for general values of r. We also provide infinite families of values r with optimal such codes and study the case of small values of r.

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On a new class of codes for identifying vertices in graphs

Cited by 377 publications

References 19 publications

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Localization and identification in networks using robust identifying codes

Tolerant identification with Euclidean balls

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