Computing minimal doubly resolving sets of graphs

Kratica, Jozef; angalović, Mirjana; Kovačevíc-Vujčić, Vera

doi:10.1016/j.cor.2008.08.002

Cited by 62 publications

(43 citation statements)

References 25 publications

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“…The metric dimension problem and minimal doubly resolving set problem are NP-hard. The proofs can be found in [6] and [8], respectively. The problem of finding minimal doubly resolving set for different families of graphs has been studied in [5] and [9].…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On Minimal Doubly Resolving Sets of Circulant Graphs

Ahmad¹,

Sultan²

2017

AMS

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Consider a simple connected undirected graph G = (V G ,E G), where V G represents the vertex set and E G represents the edge set respectively. A subset B of V G is called a resolving set if for every two distinct vertices x, y of G there is a vertex v in set B such that d(x,v) ≠ d(y,v). The resolving set of minimum cardinality is called metric basis of graph G. This minimal cardinality of metric basis is denoted by b(G), and is called metric dimension of G. A subset D of V is called doubly resolving set if for every two vertices x, y of G there are two vertices u, v ∈ D such that d(u,x)-d(u,y) ≠ d(v,x)-d(v,y). A doubly resolving set with minimum cardinality is called minimal doubly resolving set. This minimum cardinality is denoted by y(G). Some partial cases for metric dimension of circulant graph C n (1,2,3) for n ≥ 12 has been discussed in [21]. Afterwards, problem of finding metric dimension for circulant graph C n (1,2,3), n ≥ 12 has been completely solved by Borchert et al., in [7]. In this paper, we prove that

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On Minimal Doubly Resolving Sets of Circulant Graphs

Ahmad¹,

Sultan²

2017

AMS

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“…Detailed description of GA is out of this paper's scope and it can be found in [26]. Extensive computational experience on various optimization problems shows that GA often produces high quality solutions in a reasonable time, as can be seen from the following recent applications [8,19,20,21,27,32].…”

Section: Proposed Ga Methodsmentioning

confidence: 99%

A genetic algorithm for the minimum hitting set

Cendic-Lazovic¹

2014

Sci Publ State Univ Novi Pazar Ser A

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In this paper the Minimum Hitting Set (MHS) problem is considered. The problem is solved by a genetic algorithms (GA) that uses binary encoding and standard genetics operators adapted to the problem. In proposed implementation all individuals are feasibility by default, so search is directed into the promising regions. The overall performance of the GA implementation is improved by caching technique. Local search optimization is used to refine the solutions explored by GAs. The algorithm is tested on standard instances from the literature. The obtained results are also compared with the results of existing methods for solving MHS in order to assess their merits.

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“…The problem of minimizing the required number of sensors in order to identify the source in the zero-variance setting has first been studied in relation to the DRS problem by Chen et al [9]; in fact, a sensor set U such that the number of equivalence classes is q = N (and hence the source can always be identified) is nothing but a DRS. Finding a Doubly Resolving Set of minimum size is known to be NP-hard [27]. Chen et al proposed an approximation algorithm based on a greedy minimization of an entropy function [9].…”

Section: Supplementary Materialsmentioning

confidence: 99%

A General Framework for Sensor Placement in Source Localization

Spinelli

Celis

Thiran

2019

IEEE Trans. Netw. Sci. Eng.

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When an epidemic spreads in a given network of individuals or communities, can we detect its source using only the information provided by a small set of nodes? We propose a general framework that incorporates two dimensions. First, we can either rely exclusively on a set of selected nodes (i.e., sensors) which always reveal their state independently of any particular epidemic (these are called static), or we can add some sensors (called dynamic) as an epidemic spreads, depending on which additional information is required. Second, the method can either localizes the source after an epidemic has spread through the entire network (offline), or while the epidemic is ongoing (online). We empirically study the performance of offline and online localization both with and without dynamic sensors. Our analysis shows that, by using dynamic sensors, the number of sensors necessary to localize the source is reduced by up to a factor of 10 and that, even with high-variance transmission delays, the source can be localized by using fewer than 5% of the nodes as sensors.

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Computing minimal doubly resolving sets of graphs

Cited by 62 publications

References 25 publications

On Minimal Doubly Resolving Sets of Circulant Graphs

On Minimal Doubly Resolving Sets of Circulant Graphs

A genetic algorithm for the minimum hitting set

A General Framework for Sensor Placement in Source Localization

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