PTAS for Minimum k-Path Connected Vertex Cover in Growth-Bounded Graphs

The minimum sensor cover is an important optimization problem in wireless sensor networks, which can be seen as a special case of the minimum set cover problem. The minimum set cover problem is a well-known problem in combinatorial optimization, which has no polynomial-time (ρlnδ)-approximation for 0<ρ<1 unless NP⊆DTIME(n O(loglogn)) where δ is the maximum cardinality of a subset in the input collection. However, the minimum sensor cover problem has polynomial-time constant-approximations because this special case has a geometric structure. The design technique, called partition, is employed to take the advantage of this geometric structure. Especially, the double partition plays an important role. In this article, we would like to give an exploratory essay for the double partition technique together with research progress on approximations for the minimum sensor cover problem.

We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.

An Quality of Service(QOS) secure routing algorithm based on ant colony optimization is put forward to solve a variety of security problems in wireless sensor networks. The algorithm combines the ant colony optimization algorithm with the credit evaluation mechanism, and introduces the node 's credit as the control factor to obtain better security performance. The algorithm first cuts the network topology according to the QOS requirements, kicks out the nodes that do not meet the QOS requirements, and then quickly converges the route to the optimal route according to the ant colony optimization method. Finally, the security analysis of the algorithm is carried out from various aspects of network layer attack. The experimental results show that the algorithm has very good performance in wireless sensor networks. As a result, it is concluded that it can be widely applied in wireless sensor networks.

A faster FPTAS for knapsack problem with cardinality constraint

The relay node placement problem in wireless sensor network (WSN) aims at deploying the minimum number of relay nodes over the network so that each sensor can communicate with at least one relay node. When the deployed relay nodes are homogeneous and their communication ranges are circular, one way to solve the WSN relay node placement problem is to solve the minimum geometric disk cover (MGDC) problem first and place the relay nodes at the centers of the covering disks and then, if necessary, deploy additional relay nodes to meet the connection requirement of relay nodes. It is known that the MGDC problem is NP-complete. A novel linear time approximation algorithm for the MGDC problem is proposed, which identifies covering disks using the regular hexagon tessellation of the plane with bounded area. The approximation ratio of the proposed algorithm is (5+ϵ), where0&lt;ϵ≤15. Experimental results show that the worst case is rare, and on average the proposed algorithm uses less than 1.7 times the optimal disks of the MGDC problem. In cases where quick deployment is necessary, this study provides a fast 7-approximation algorithm which uses on average less than twice the optimal number of relay nodes in the simulation.

In a wireless sensor network, the size of radio nodes has direct relation to the cost of total wireless sensor networks, and at the same time, the problem is closely connected to wireless sensor networks' performance, such as robust, fault-tolerance, and further more, it is considered at first as wireless sensor networks are designed. Therefore, the research on the size of radio nodes has significant meanings of theory and practice to design of wireless sensor networks. The paper simplifies complex coverage problem step by step. By means of math modeling, theoretical analysis and formula deducting, classical geometric theories and the method of mathematics induction are adopted, at last, the analysis formula of minimum number of nodes is theoretically educed under circumstances of entire and seamless coverage in WSN. In addition, it is worth mentioning that the conclusion is the full same with computing the number of communication nodes of wireless sensor networks.

#NFA refers to the problem of counting the words of length n accepted by a non-deterministic finite automaton. #NFA is #P-hard, and although fully-polynomial-time randomized approximation schemes (FPRAS) exist, they are all impractical. The first FPRAS for #NFA had a running time of Õ(n 17 m 17 ε -14 łog(δ -1 )), where m is the number of states in the automaton, δ ∈ (0,1] is the confidence parameter, and ε &gt; 0 is the tolerance parameter (typically smaller than 1). The current best FPRAS achieved a significant improvement in the time complexity relative to the first FPRAS and obtained FPRAS with time complexity Õ((n 10 m 2 + n 6 m 3 )ε -4 łog 2 (δ -1 )). The complexity of the improved FPRAS is still too intimidating to attempt any practical implementation. In this paper, we pursue the quest for practical FPRAS for #NFA by presenting a new algorithm with a time complexity of O(n 2 m 3 łog(nm)ε -2 łog(δ -1 )). Observe that evaluating whether a word of length n is accepted by an NFA has a time complexity of O(nm 2 ). Therefore, our proposed FPRAS achieves sub-quadratic complexity with respect to membership checks.

PurposeThe purpose of this paper is to find the best approximation algorithm for solving the more general case of single‐supplier multi‐retailer capacitated economic lot‐sizing (SM‐CELS) problem in deterministic inventory theory, which is the non‐deterministic polynomial (NP)‐hard problem.Design/methodology/approachSince few theoretical results have been published on polynomial time approximation algorithms for SM‐CELS problems, this paper develops a fully polynomial time approximation scheme (FPTAS) for the problem with monotone production and holding‐backlogging cost functions. First the optimal solution of a rounded problem is presented as the approximate solution and its straightforward dynamic‐programming (DP) algorithm. Then the straightforward DP algorithm is converted into an FPTAS by exploiting combinatorial properties of the recursive function.FindingsAn FPTAS is designed for the SM‐CELS problem with monotone cost functions, which is the strongest polynomial time approximation result.Research limitations/implicationsThe main limitation is that the supplier only manufactures without holding any products when the model is applied.Practical implicationsThe paper presents the best result for the SM‐CELS problem in deterministic inventory theory.Originality/valueThe LP‐rounding technique, an effective approach to design approximation algorithms for NP‐hard problems, is successfully applied to the SM‐CELS problem in this paper.

We study the integer maximum flow problem on wireless sensor networks with energy constraint. In this problem, sensor nodes gather data and then relay them to a base station, before they run out of battery power. Packets are considered as integral units and not splittable. The problem is to find the maximum data flow in the sensor network subject to the energy constraint of the sensors. We show that this integral version of the problem is strongly NP-complete and in fact APX-hard. It follows that the problem is unlikely to have a polynomial time approximation scheme. Even when restricted to graphs with concrete geometrically defined connectivity and transmission costs, the problem is still strongly NP-complete. We provide some interesting polynomial time algorithms that give good approximations for the general case nonetheless. For networks of bounded treewidth greater than two, we show that the problem is weakly NP-complete and provide pseudo-polynomial time algorithms. For a special case of graphs with treewidth two, we give a polynomial time algorithm.

A faster FPTAS for counting two-rowed contingency tables

The coverage problem in wireless sensor networks is to place sensor devices in a service area so that the entire service area is covered/sensed. We have modeled the coverage problem as two sub-problems: floorplan and placement. The floorplan problem is to partition the service area into well-defined geometric cells, whereas the placement problem is to assign the sensor devices into a set of cells. The cardinality of the set of cells is considered as all possible candidate locations to place the sensors; therefore, we transformed the search space from continuous to discrete. Thus, we have reduced the complexity of the coverage problem, but we have merged the two sub-problems (floorplan and placement) into a single optimization problem to achieve good solutions. The objective function is to maximize the coverage of the service area while not exceeding a given budget. The merged optimization problem has been coded into the evolutionary approach. We have conducted five studies, where three studies compare the behavior of coverage ratio, cost, and radius of coverage of all allocated sensors with the number of generations separately. The fourth study examines the cell size versus coverage ratio. The fifth study looks at the average number of neighbors covered by a sensor with respect to the generations. Our experimental results present interesting findings in all five studies and our computerized evolutionary methodology generates solutions under 5 minutes.

Barrier coverage, as one of the most important applications of wireless sensor network (WSNs), is to provide coverage for the boundary of a target region. We study the barrier coverage problem by using a set of n sensors with adjustable coverage radii deployed along a line interval or circle. Our goal is to determine a range assignment $$\mathbf {R}=({r_{1}},{r_{2}}, \ldots , {r_{n}})$$ of sensors such that the line interval or circle is fully covered and its total cost $$C(\mathbf {R})=\sum _{i=1}^n {r_{i}}^\alpha $$ is minimized. For the line interval case, we formulate the barrier coverage problem of line-based offsets deployment, and present two approximation algorithms to solve it. One is an approximation algorithm of ratio 4 / 3 runs in $$O(n^{2})$$ time, while the other is a fully polynomial time approximation scheme (FPTAS) of computational complexity $$O(\frac{n^{2}}{\epsilon })$$ . For the circle case, we optimally solve it when $$\alpha = 1$$ and present a $$2(\frac{\pi }{2})^\alpha $$ -approximation algorithm when $$\alpha > 1$$ . Besides, we propose an integer linear programming (ILP) to minimize the total cost of the barrier coverage problem such that each point of the line interval is covered by at least k sensors.

Coverage in wireless sensor networks is a critical issue since it reflects the cost and detection capability of the network. Sensor coverage model is closely to the sensing functions of different types of sensors while network coverage is a collaborative measure from geographically distributed sensors. The coverage problems in wireless sensor networks have been formulated in various ways with different assumptions, constraints and objectives. In this work, a collaborative coverage model is proposed in the target surveillance application, while sensor sensing probabilistic model is introduced. Furthermore, a collaborative target coverage optimization algorithm (CTCOMSN) is presented to prolong the network lifetime in energy constraint mobile sensor networks. Simulation results show that CTCOMSN can prolong the lifetime greatly while improving the sensing quality.

The activity scheduling of sensors to alternately wake up for sensing obligation such that the network lifetime can be efficiently prolonged is a very important issue in wireless sensor networks (WSNs). Target area coverage is a new coverage problem in wireless sensor networks. The paper addresses the target area coverage problem and schedules sensors to alternatively wake up to collaboratively cover and sense the target area. A geometric-based activity scheduling scheme, named GAS scheme, for WSNs to fully cover a target area is proposed. By means of computational geometry, the sensors can self-determine when to sleep or wake up while preserving the sensing coverage. GAS is able to find as few sensors as possible to cover the target area, which is termed a cover set. In addition, GAS can find as many number of cover sets as possible to be alternately in charge of the sensing task. Simulation results show that GAS can efficiently schedule the sensor when to switch between active and sleep modes. Furthermore, the network lifetime can be prolonged significantly in comparison with the state-of-the-art schemes.

Coverage is a typical problem in wireless sensor networks to fulfil the issued sensing tasks. In general, sensing coverage represents how well an area is monitored by sensors. The quality of a sensor network can be reflected by the levels of coverage and connectivity that it offers. The coverage problem has been studied extensively, especially when combined with connectivity and energy efficiency. Constructing a connected fully covered, and energy efficient sensor network is valuable for real world applications due to the limited resources of sensor nodes. In this paper, we survey recent contributions addressing energyefficient coverage problems in the context of static WASNs, networks in which sensor nodes do not move once they are deployed and present in some detail of the algorithms, assumptions, and results. A comprehensive comparison among these approaches is given from the perspective of design objectives, assumptions, algorithm attributes and related results.