+\section{Centralized Algorithms}
+\label{ch2:sec:02}
+The major idea of most centralized algorithms is to divide/organize the sensors into a suitable number of cover sets and to activate these cover sets successively. The centralized algorithms always provide optimal or near-optimal solution since the algorithm has a global view of the whole network. Energy-efficient centralized approaches differ according to several criteria \cite{ref113}, such as the coverage objective (target coverage or area coverage), the node deployment method (random or deterministic), and the heterogeneity of sensor nodes (common sensing range, common battery lifetime).
+
+The first algorithms proposed in the literature consider that the cover sets are disjoint: a sensor node appears in exactly one of the generated cover sets~\cite{ref114,ref115,ref116,ref227}. For instance, Slijepcevic and Potkonjak \cite{ref116} propose an algorithm, which allocates sensor nodes in mutually independent sets to monitor an area divided into several fields. Their algorithm builds a cover set by including in priority the sensor nodes which cover critical fields, that is to say, fields that are covered by the smallest number of sensors. The time complexity of their heuristic is $O(n^2)$ where $n$ is the number of sensors.
+%%%M. Cardei et al.~\cite{ref227}, suggest a graph coloring technique to achieve energy savings by organizing the sensor nodes into a maximum number of disjoint dominating sets, which are activated successively. They have defined the maximum disjoint dominating sets problem and they have produced a heuristic that computes the disjoint cover sets so as to monitor the area of interest. The dominating sets do not guarantee the coverage of the whole region of interest. Abrams et al.~\cite{ref114} design three approximation algorithms for a variation of the set k-cover problem, where the objective is to partition the sensors into covers so that the number of covers that include an area, summed over all areas, is maximized. Their work is built upon previous work in~\cite{ref116} and the generated cover sets do not provide complete coverage of the monitoring zone.
+%%%The authors in~\cite{ref115} propose a heuristic to compute the Disjoint Set Covers (DSC). In order to compute the maximum number of covers, they first transform DSC into a maximum-flow problem, which is then formulated as a Mixed Integer Programming problem (MIP). Based on the solution of the MIP, they design a heuristic to compute the final number of covers. The results show a slight performance improvement in terms of the number of produced DSC in comparison to~\cite{ref116}, but it incurs higher execution time due to the complexity of the mixed integer programming solving. Zorbas et al. \cite{ref228} present B\{GOP\}, a centralized target coverage algorithm introducing sensor candidate categorization depending on their coverage status and the notion of critical target to call targets that are associated with a small number of sensors. The total running time of their heuristic is $0(m n^2)$, where $n$ is the number of sensors and $m$ the number of targets. Compared to algorithm's results of Slijepcevic and Potkonjak \cite{ref116}, their heuristic produces more cover sets with a slight growth rate in execution time.
+L. Liu et al.~\cite{ref150} formulate the maximum disjoint sets for maintaining target coverage and connectivity problem in WSN. They propose a graph theoretical framework to study the joint problem of connectivity and coverage in a WSN with random deployment of nodes with no restrictions on the sensing and communication ranges of nodes. They propose heuristic algorithms to find the suitable number of nodes to guarantee connectivity and coverage while maximizing network lifetime.
+Y. Li et al.~\cite{ref142} present a framework with heuristic strategies to solve the area coverage problem. The framework converts any complete coverage problem into a partial coverage one with any coverage ratio. They execute a complete coverage algorithm to find full coverage sets with virtual radii and then transform the coverage sets to a partial coverage sets by adjusting sensing radii. This framework has four strategies, two of them are designed for network, where the sensors have fixed sensing range and the other two are for network, where the sensors have adjustable sensing range. The properties of the algorithms can be maintained by this framework and the transformation process has a low execution time. The simulation results validate the efficiency of the four proposed strategies. More recently, Deschinkel and Hakem \cite{ref229} introduce a near-optimal heuristic algorithm for solving the target coverage problem in WSN. The sensor nodes are organized into disjoint cover sets, each one is capable of monitoring all the targets of the region of interest. Their algorithm is able to construct the different cover sets in parallel. The results show that their algorithm achieves near-optimal solutions compared to the optimal ones obtained by the resolution of an integer programming problem.
+%exact method.
+
+
+In the case of non-disjoint algorithms~\cite{ref117,ref167,ref144,ref147,ref118}, sensors may participate in more than one cover set. In some cases, this may prolong the lifetime of the network in comparison to the disjoint cover set algorithms, but designing algorithms for non-disjoint cover sets generally induces a higher order of complexity. Moreover, in case of a sensor's failure, non-disjoint scheduling policies are less resilient and reliable because a sensor may be involved in more than one cover sets. For instance,
+%%%Cardei et al.~\cite{ref167} present a Linear Programming (LP) solution and a greedy approach to extend the sensor network lifetime by organizing the sensors into a maximal number of non-disjoint cover sets. Simulation results show that by allowing sensors to participate in multiple sets, the network lifetime increases compared with related work~\cite{ref115}.
+the authors in~\cite{ref148}, address the problem of minimum cost area coverage in which full coverage is performed by using the minimum number of sensors for an arbitrary geometric shape region. A geometric solution to the minimum cost coverage problem under a deterministic deployment is proposed. The probabilistic coverage solution which provides a relationship between the probability of coverage and the number of randomly deployed sensors in an arbitrarily-shaped region is suggested.
+%%%The work in~\cite{ref144} addresses the area coverage problem by proposing a Geometrically based Activity Scheduling scheme, named GAS, to fully cover the area of interest in WSNs. The authors deal with a small area, called target area coverage, which can be monitored by a single sensor instead of area coverage, which focuses on a large area that should be monitored by many sensors cooperatively. They explain that GAS is capable to monitor the target area by using the fewest number of sensors and it can produce as many cover sets as possible. A novel area coverage method to divide the sensors called Node Coverage Grouping (NCG) is suggested~\cite{ref147}. The sensors in the connectivity group are within sensing range of each other and the data collected by those in the same group are supposed to be similar. They prove that dividing N sensors via NCG into connectivity groups is an NP-hard problem. So, a heuristic algorithm of NCG with time complexity of $O(n^3)$ is proposed. For some applications, such as monitoring an ecosystem with extremely diversified environment, it might be a premature assumption that sensors near to each other sense similar data.
+The problem of k-coverage over the area of interest in WSNs is addressed in~\cite{ref152}. It is mathematically formulated and the spatial sensor density for full k-coverage is determined. The relation between the communication range and the sensing range is constructed by this work to retain the k-coverage and connectivity in WSN. After that, four configuration protocols are proposed for treating the k-coverage in WSNs. Simulation results show that their protocols outperform an existing distributed k-coverage configuration protocol. The work presented in~\cite{ref151} solves the area coverage and connectivity problem in sensor networks in an integrated way. The network lifetime is divided into a fixed number of rounds. A coverage bitmap of sensors of the domain is generated in each round and based on this bitmap, it is decided which sensors stay active or go to sleep. They check the connection of the graph via laplacian of the adjacency graph of active sensors in each round. They define the connected coverage problem as an optimization problem and a centralized genetic algorithm is used to find the solution. Recent studies show an increasing interest in the use of exact schemes to solve optimization problems in WSNs \cite{ref230,ref231,ref121,ref122,ref120}. Column Generation (CG) has been widely used to address different versions of Maximum-network Lifetime Problem (MLP). CG decomposes the problem into a Restricted Master Problem (RMP) and a Pricing Subproblem (PS). The former maximizes lifetime using an incomplete set of columns and the latter is used to identify new profitable columns.
+%%%A. Rossi et al.~\cite{ref121} introduce an efficient implementation of a genetic algorithm based on CG to extend the lifetime and maximize target coverage in wireless sensor networks under bandwidth constraints. The authors show that the use of metaheuristic methods to solve PS in the context of CG allows to obtain optimal solutions quite fast and to produce high-quality solutions when the algorithm is stopped before returning an optimal solution. More recently,
+ F. Castano et al. \cite{ref120} address the maximum network lifetime and the target coverage problem in WSNs with connectivity and coverage constraints. They consider two cases to schedule the activity of a set of sensor nodes, keeping them connected while network lifetime is maximized. First, the full coverage of the targets is required, second only a fraction of the targets has to be monitored at any instant of time. They propose an exact approach based on column generation boosted by a Greedy Randomized Adaptive Search Procedure (GRASP) and a Variable Neighborhood Search (VNS) to address both of these problems. Finally, a multiphase framework combining these two approaches is constructed sequentially using these two heuristics at each iteration of the column generation algorithm. The results show that combining the two heuristic methods enhances the results significantly.
+
+More recently,
+%%%the authors in~\cite{ref118} consider an area coverage optimization algorithm based on linear programming approach to select the minimum number of working sensor nodes, in order to preserve a maximum coverage and to extend the lifetime of the network. The experimental results show that linear programming can provide a fewest number of active nodes and maximize the network lifetime coverage.
+M. Rebai et al.~\cite{ref141}, formulate the problem of full grid area coverage problem using two integer linear programming models: the first, is a model that takes into account only the overall coverage constraint; the second, both the connectivity and the full grid coverage constraints are taken into consideration. This work does not consider the energy constraint. H. Cheng et al.~\cite{ref119} define a heuristic area coverage algorithm called Cover Sets Balance (CSB), which chooses a set of active nodes using the tuple (data coverage range, residual energy). Then, they introduce a new Correlated Node Set Computing (CNSC) algorithm to find the correlated node set for a given node. After that, they propose a High Residual Energy First (HREF) node selection algorithm to minimize the number of active nodes. X. Liu et al.~\cite{ref143} explain that in some applications of WSNs such as Structural Health Monitoring (SHM) and volcano monitoring, the traditional coverage model which is a geographic area defined for individual sensors is not always valid. For this reason, they define a generalized area coverage model, which is not required to have the coverage area of individual nodes, but only based on a function deciding whether a set of sensor nodes is capable of satisfy the requested monitoring task for a certain area. They propose two approaches for dividing the deployed nodes into suitable cover sets.
+
+
+
+%Various centralized methods based on column generation approaches have also been proposed in~\cite{ref120,ref121,ref122}.
