One model for viral marketing is the positive influence. In this model, an inactive node is changed into active if and only if at least half of its neighbors are already in active state. The positive influence model can be viewed as a special case of a general threshold model, in which the threshold function at each node has value one if at least a certain fraction of neighbors are in active state, and value 0 otherwise. This function can be proved to be monotonically increasing and nonsubmodular for any predefined fraction. Therefore, given a seed set, the number of influenced nodes is not submodular with respect to the size of the seed set. This fact makes those optimization problems related with positive influence very hard, including the minimum partial positive influence seeding problem: Given a social network G = (V, E) and a number 02H ([pn]))-approximation algorithm for the minimum partial positive influence seeding problem, where n is the number of nodes, and H(·) is the Harmonic number.

Energy efficiency is an important issue in the study of wireless sensor networks. Given a set of targets and a set of sensors with bounded lifetime, the maximum lifetime k-coverage problem is to schedule active/sleeping status of sensors to maximize the time period during which every target is covered by at least k active sensors. Previously, it was known that when the sensing ranges are uniform, this problem has a polynomial time (4+ε)-approximation for k = 1 and (6+ε)-approximation for k = 2. In this paper, we make significant progress by showing that for any positive integer k, there exists a polynomial-time (3 + ε)-approximation.

Energy efficiency is an important issue in the study of wireless sensor networks. Given a homogeneous set of sensors with unit lifetime and a set of target points, find an active/sleeping schedule for sensors to maximize the lifetime of k-coverage, i.e., the time period during which every target point is covered by at least k active sensors. This is a well known problem in wireless sensor networks concerning with energy efficiency. When k = 1, it is called the maximum lifetime coverage problem which has been proved to have a polynomial-time (4 + ε)-approximation. When k ≥ 2, it is the maximum lifetime fault-tolerant coverage problem. Previous to this work, only in the case k = 2, a polynomial-time (6 + ε)-approximation is found. In this paper, we will make a significant progress by showing that for any positive integer k, there exists a polynomial-time (4 + ε)-approximation, and for k = 1,2, the performance ratio can be improved to (3 + ε).

One type of distributed systems is the client/server system consist of clients and servers. In order to improve the performance of such a system, client assignment strategy plays an important role. There are two criteria to evaluate the load on the servers - total load and load balance. The total load increases when the load balance decreases, vice versa. It has been proved that finding the best client assignment is NP-hard. In this paper, we propose a new model for the client assignment problem and design an algorithm based on Semidefinite programming (SDP). Our method has a (relaxed) performance ratio 0.87 when only 2 servers exist. In general case, our method becomes a heuristic, and the ratio of each iteration is 0.87. We are the first one to give these bounds. Our simulation results are compared with the state-of-art client assignment method, and our strategy outperforms it in terms of running time while keeps the load in similar level.

When a large amount of sensors are randomly deployed into a field, how can we make a sleep/activate schedule for sensors to maximize the lifetime of target coverage in the field? This is a well-known problem, called Maximum Lifetime Coverage Problem (MLCP), which has been studied extensively in the literature. It is a long-standing open problem whether MLCP has a polynomial-time constant-approximation. The best-known approximation algorithm has performance ratio 1 + ln n where n is the number of sensors in the network, which was given by Berman et. al [1]. In their work, MLCP is reduced to Minimum Weight Sensor Coverage Problem (MWSCP) which is to find the minimum total weight of sensors to cover a given area or a given set of targets with a given set of weighted sensors. In this paper, we present a polynomial-time (4 + ∈)-approximation algorithm for MWSCP and hence we obtain a polynomial-time (4 + ξ)-approximation algorithm for MLCP, where ∈ >; 0, ξ >; 0.

Directional antennas can divide the transmission range into several sectors. Thus, through switching off sectors in unnecessary directions in wireless networks, we can save bandwidth and energy consumption. In this paper, we will study a directional virtual backbone (VB) in the network where directional antennas are used. When constructing a VB, we will take routing and broadcasting into account since they are two common operations in wireless networks. Hence, we will study a VB with guaranteed routing costs, named α Minimum rOuting Cost Directional VB (α-MOC-DVB). Besides the properties of regular VBs, α-MOC-DVB also has a special constraint - for any pair of nodes, there exists at least one path all intermediate directions on which must belong to α-MOC-DVB and the number of intermediate directions on the path is smaller than α times that on the shortest path. We prove that construction of a minimum α-MOC-DVB is an NP-hard problem in a general directed graph. A heuristic algorithm is proposed and theoretical analysis is also discussed in the paper. Extensive simulations demonstrate that our α-MOC-DVB is much more efficient in the sense of VB size and routing costs compared to other VBs.

It is well-known that the application of directional antennas can help conserve bandwidth and energy consumption in wireless networks. Thus, to achieve efficiency in wireless networks, we study a special virtual backbone (VB) using directional antennas, requiring that from one node to any other node in the network, there exists at least one directional shortest path all of whose intermediate directions should belong to the VB, named as Minimum rOuting Cost Directional VB (MOC-DVB). In addition, VB has been well studied in Unit Disk Graph (UDG). However, radio wave based communications in wireless networks may be interrupted by obstacles (e.g., buildings and mountains). Thus, in this paper, we model a network as a general directed graph. We prove that construction of a minimum MOC-DVB is an NP-hard problem in a general directed graph and in term of the size of MOC-DVB, there exists an unreachable lower bound of the polynomial-time selected MOC-DVB. Therefore, we propose a distributed approximation algorithm for constructing MOC-DVB with approximation ratio of 1 + ln K + 2ln δD, where K is the number of antennas on each node and δDis the maximum direction degree in the network. Extensive simulations demonstrate that our constructed MOC-DVB is much more efficient in the sense of MOC-DVB size and routing cost compared to other VBs.

Topology control is one vital factor to a wireless network's efficiency. A Connected Dominating Set (CDS) can be a useful basis of a backbone topology construction. In this paper, a special CDS, named \alpha Minimum rOuting Cost CDS (\alpha-MOC-CDS), will be studied to improve the performance of CDS based broadcasting and routing. In this paper, we prove that construction of a minimum \alpha-MOC-CDS is NP-hard in a general graph and we propose a heuristic algorithm for construction of \alpha-MOC-CDS.

How to adjust the transmission power at each node to achieve global energy efficiency while maintaining the network connectivity, referred as power management problem, is the major target of various topology control technologies. Moreover, fault tolerance which is often modeled as 2-edge or 2-vertex connectivity is another desired feature in many applications. In this paper, we study the fault tolerant dual power assignment problem. With the assumption of dual universal transmission power levels, we aim to minimize the total number of nodes assigned to high power level such that the resultant network topology is 2-edge or 2-vertex connected. As the problems are NP-hard, we design a novel algorithm to compute nearly-optimal solutions. From the theoretical perspective, we prove that our algorithm can guarantee 3.67-approximation for both 2-edge connectivity and 2-vertex connectivity, which improves the existing best approximation algorithm. We also conduct some numerical experiments which show that results of our algorithm are at most 2 times of optimal solutions in average and have significant improvements compared to that of existing algorithm.

Unlike a cellular or wired network, there is no base station or network infrastructure in a wireless ad-hoc network, in which nodes communicate with each other via peer communications. In order to make routing and flooding efficient in such an infrastructureless network, Connected Dominating Set (CDS) as a virtual backbone has been extensively studied. Most of the existing studies on the CDS problem have focused on unit disk graphs, where every node in a network has the same transmission range. However, nodes may have different powers due to difference in functionalities, power control, topology control, and so on. In this case, it is desirable to model such a network as a disk graph where each node has different transmission range. In this paper, we define Minimum Strongly Connected Dominating and Absorbent Set (MSCDAS) in a disk graph, which is the counterpart of minimum CDS in unit disk graph. We propose a constant approximation algorithm when the ratio of the maximum to the minimum in transmission range is bounded. We also present two heuristics and compare the performances of the proposed schemes through simulation.