+Several algorithms to retain the coverage and maximize the network lifetime were proposed in~\cite{ref113,ref101,ref103,ref105}. Table~\ref{Table1:ch2} summarized the main characteristics of some coverage approaches in previous literatures. In table~\ref{Table1:ch2}, the "SET K-COVER" characteristic refers to the maximum number of disjoint or non-disjoint sets of sensors such that each set cover can assure the coverage for the whole region. The k-coverage characteristic refers to that every point inside the monitored area is always covered by at least k active sensors.
+
+
+
+\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, where each set completely covers an interest region 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}. 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}, have suggested 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} designed three approximation algorithms for a variation of the set k-cover problem, where the objective is to partition the sensors into covers such that the number of covers that include an area, summed over all areas, is maximized.
+Their work builds 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} proposed 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} presented 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} formulated the maximum disjoint sets for maintaining target coverage and connectivity (MDS-MCC) problem in WSN. Two algorithms are proposed for solving this problem, the heuristic algorithm and network flow algorithm.
+%This work did not take into account the sensor node failure, which is an unpredictable event because the two solutions are full centralized algorithms.
+Y. Li et al.~\cite{ref142} presented a framework with heuristic strategies to solve the area coverage problem. The framework converts any complete coverage problem to a partial coverage one with any coverage ratio by means of executing a complete coverage algorithm to find a full coverage sets with virtual radii and transforming the coverage sets to a partial coverage sets by adjusting sensing radii . This framework has four strategies, two of them are designed for the network where the sensors have fixed sensing range and the other two are for the network where the sensors have adjustable sensing range. The properties of the original 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{229} introduced a near-optimal heuristic algorithm for solving the target coverage problem in WSN. The sensor nodes are organized into disjoint cover sets by the resolution an integer programming problem. Each cover set is capable of monitoring all the targets of the region of interest. Those covers sets are scheduled periodically. Their algorithm 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 exact method.
+
+
+
+In the case of non-disjoint algorithms~\cite{ref117}, 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}, addressed 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 authors are explained that with a random deployment about seven times more nodes are required to supply full coverage. The work in~\cite{ref144} addressed 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 explained that GAS is capable to monitor the target area by using a few sensors as possible and it can produce as many cover sets as possible. A novel area coverage method to divide the sensors of the WSN is 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 them in the same group are supposed to be similar. They are proved 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 premature assumption that sensors near to each other sense similar data. The problem of k-coverage over the area of interest in WSNs was 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 have proposed for treating the k-coverage in WSNs. Simulation results show that their protocols outperform an existing distributed k-coverage configuration protocol. The work that presented in~\cite{ref151} solved 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 has been generated in each round and based on this bitmap, it has been decided which sensors stay active or turn it to sleep. They checked the connection of the graph via laplacian of the adjacency graph of active sensors in each round. The generation of coverage bitmap by using Minkowski technique, the network is able to providing the desired ratio of coverage. They defined 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 WSN \cite{ref230,ref231,ref121,ref122,ref120}. Column Generation (CG) has been widely used to address different versions of 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} introduced an efficient implementation of a genetic algorithm based 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} addressed the maximum network lifetime and the target coverage problem in WSNs with connectivity and coverage constraints. They considered 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, and second only a fraction of the targets has to be monitored at any instant of time. They proposed an exact approach based on column generation and 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 by 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.
