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Hardness results and approximation algorithms for identifying codes and locatingdominating codes in graphs. Algorithmic Operations Research 3(1):43–50
, 2008
"... In a graph G = (V, E), an identifying code of G (resp. a locatingdominating code of G) is a subset of vertices C ⊆ V such that N [v]∩C 6 = ∅ for all v ∈ V, and N [u] ∩C 6 = N [v]∩C for all u 6 = v, u, v ∈ V (resp. u, v ∈ V r C), where N [u] denotes the closed neighbourhood of v, that is N [u] = N ..."
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In a graph G = (V, E), an identifying code of G (resp. a locatingdominating code of G) is a subset of vertices C ⊆ V such that N [v]∩C 6 = ∅ for all v ∈ V, and N [u] ∩C 6 = N [v]∩C for all u 6 = v, u, v ∈ V (resp. u, v ∈ V r C), where N [u] denotes the closed neighbourhood of v, that is N [u] = N(u) ∪ {u}. These codes model faultdetection problems in multiprocessor systems and are also used for designing locationdetection schemes in wireless sensor networks. We give here simple reductions which improve results of the paper [I. Charon, O. Hudry, A. Lobstein, Minimizing the Size of an Identifying or LocatingDominating Code in a Graph is NPhard, Theoretical Computer Science 290(3) (2003), 2109–2120], and we show that minimizing the size of an identifying code or a locatingdominating code in a graph is APXhard, even when restricted to graphs of bounded degree. Additionally, we give approximation algorithms for both problems with approximation ratio O(ln V ) for general graphs and O(1) in the case where the degree of the graph is bounded by a constant. Key words: approximation algorithms, approximation hardness, identifying codes, locatingdominating codes, fault tolerance, domination problems, combinatorial optimization, graph algorithms.
Identifying Codes and the Set Cover Problem
, 2006
"... We consider the problem of finding a minimum identifying code in a graph, i.e., a designated set of vertices whose neighborhoods uniquely overlap at any vertex on the graph. This identifying code problem was initially introduced in 1998 and has been since fundamentally connected to a wide range of a ..."
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Cited by 12 (4 self)
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We consider the problem of finding a minimum identifying code in a graph, i.e., a designated set of vertices whose neighborhoods uniquely overlap at any vertex on the graph. This identifying code problem was initially introduced in 1998 and has been since fundamentally connected to a wide range of applications, including fault diagnosis, location detection, environmental monitoring, and connections to information theory, superimposed codes, and tilings. Though this problem is NPcomplete, its known reduction is from 3SAT and does not readily yield an approximation algorithm. In this paper we show that the identifying code problem is computationally equivalent to the set cover problem and present a Θ(log n)approximation algorithm based on the greedy approach for set cover; we further show that, subject to reasonable assumptions, no polynomialtime approximation algorithm can do better. Finally, we show that a generalization of the identifying codes problem, for which no complexity results were known thusfar, is NPhard. 1
Identifying codes and covering problems
 IEEE Transaction on Information Theory
, 2008
"... 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 ..."
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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 wellknown setcovering problem, resulting in a tight hardness of approximation result and novel, provably tight polynomialtime approximations. The main results are also extended to rrobust identifying codes and analogous set (2r + 1)multicover problems. Finally, empirical support is provided for the effectiveness of the proposed approximations, including good constructions for wellknown topologies such as infinite twodimensional grids.
Disjoint identifyingcodes for arbitrary graphs
 In Proc. International Symposium on Information Theory (ISIT
, 2005
"... Identifying codes have been used in a variety of applications, including sensorbased location detection in harsh environments. The sensors used in such applications are typically battery powered making energy conservation an important optimization criterion for lengthening network lifetime. In this ..."
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Identifying codes have been used in a variety of applications, including sensorbased location detection in harsh environments. The sensors used in such applications are typically battery powered making energy conservation an important optimization criterion for lengthening network lifetime. In this work we propose and develop the concept of disjoint identifying codes with the motivation of providing energy loadbalancing in such systems. We also provide informationtheoretic upper and lower bounds on the number of disjoint identifying codes in a given graph, and show that these bounds are asymptotically tight for a modification of Hadamard matrices. A version of this paper should be presented at the IEEE Symposium on Information on Information Theory 2005. I.
Adaptive Identification in Torii in the King Lattice
"... Given a connected graph G = (V,E), Let r ≥ 1 be an integer and Br(v) denote the ball of radius r centered at v ∈ V, i.e., the set of all vertices within distance r from v. A subset of vertices C ⊆ V is an ridentifying code of G (for a given nonzero constant r ∈ N) if and only if all the sets Br(v) ..."
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Cited by 1 (1 self)
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Given a connected graph G = (V,E), Let r ≥ 1 be an integer and Br(v) denote the ball of radius r centered at v ∈ V, i.e., the set of all vertices within distance r from v. A subset of vertices C ⊆ V is an ridentifying code of G (for a given nonzero constant r ∈ N) if and only if all the sets Br(v) ∩ C are nonempty and pairwise distinct. These codes were introduced in [7] to model a faultdetection problem in multiprocessor systems. They are also used to devise locationdetection schemes in the framework of wireless sensor networks. These codes enable one to locate a malfunctioning device in these networks, provided one scans all the vertices of the code. We study here an adaptive version of identifying codes, which enables to perform tests dynamically. The main feature of such codes is that they may require significantly fewer tests, compared to usual static identifying codes. In this paper we study adaptive identifying codes in torii in the king lattice. In this framework, adaptive identification can be closely related to a Rényitype search problem studied by M. Ruszinkó [11].
Algorithms and Complexity for Metric Dimension and LocationDomination on Interval and Permutation Graphs
"... We study the problems LocatingDominating Set and Metric Dimension, which consist of determining a minimumsize set of vertices that distinguishes the vertices of a graph using either neighbourhoods or distances. We consider these problems when restricted to interval graphs and permutation graphs. ..."
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We study the problems LocatingDominating Set and Metric Dimension, which consist of determining a minimumsize set of vertices that distinguishes the vertices of a graph using either neighbourhoods or distances. We consider these problems when restricted to interval graphs and permutation graphs. We prove that both decision problems are NPcomplete, even for graphs that are at the same time interval graphs and permutation graphs and have diameter 2. While LocatingDominating Set parameterized by solution size is trivially fixedparametertractable, it is known thatMetric Dimension isW [2]hard. We show that for interval graphs, this parameterization of Metric Dimension is fixedparametertractable.
Identifying codes in vertextransitive graphs and strongly regular graphs
"... We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(V ) + 1 where V is the set of vertices of the graph. We focus on vertextransitive graphs for which we can ..."
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We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(V ) + 1 where V is the set of vertices of the graph. We focus on vertextransitive graphs for which we can compute the exact fractional solution. There are known examples of vertextransitive graphs that reach both bounds. We exhibit infinite families of vertextransitive graphs with integer and fractional identifying codes of order V α with α ∈ {14, 13, 25}. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.
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"... Identifying codes in (random) geometric networks* Tobias M"uller # JeanS'ebastien Sereni## ..."
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Identifying codes in (random) geometric networks* Tobias M&quot;uller # JeanS'ebastien Sereni##