The Online Disjoint Set Cover Problem and its Applications

Given a universe $U$ of $n$ elements and a collection of subsets $\mathcal{S}$ of $U$, the maximum disjoint set cover problem (DSCP) is to partition $\mathcal{S}$ into as many set covers as possible, where a set cover is defined as a collection of subsets whose union is $U$. We consider the online DSCP, in which the subsets arrive one by one (possibly in an order chosen by an adversary), and must be irrevocably assigned to some partition on arrival with the objective of minimizing the competitive ratio. The competitive ratio of an online DSCP algorithm $A$ is defined as the maximum ratio of the number of disjoint set covers obtained by the optimal offline algorithm to the number of disjoint set covers obtained by $A$ across all inputs. We propose an online algorithm for solving the DSCP with competitive ratio $\ln n$. We then show a lower bound of $\Omega(\sqrt{\ln n})$ on the competitive ratio for any online DSCP algorithm. The online disjoint set cover problem has wide ranging applications in practice, including the online crowd-sourcing problem, the online coverage lifetime maximization problem in wireless sensor networks, and in online resource allocation problems.Comment: To appear in IEEE INFOCOM 201

Approximation Algorithm for Line Segment Coverage for Wireless Sensor Network

The coverage problem in wireless sensor networks deals with the problem of covering a region or parts of it with sensors. In this paper, we address the problem of covering a set of line segments in sensor networks. A line segment ` is said to be covered if it intersects the sensing regions of at least one sensor distributed in that region. We show that the problem of finding the minimum number of sensors needed to cover each member in a given set of line segments in a rectangular area is NP-hard. Next, we propose a constant factor approximation algorithm for the problem of covering a set of axis-parallel line segments. We also show that a PTAS exists for this problem.Comment: 16 pages, 5 figures

Set It and Forget It: Approximating the Set Once Strip Cover Problem

We consider the Set Once Strip Cover problem, in which n wireless sensors are deployed over a one-dimensional region. Each sensor has a fixed battery that drains in inverse proportion to a radius that can be set just once, but activated at any time. The problem is to find an assignment of radii and activation times that maximizes the length of time during which the entire region is covered. We show that this problem is NP-hard. Second, we show that RoundRobin, the algorithm in which the sensors simply take turns covering the entire region, has a tight approximation guarantee of 3/2 in both Set Once Strip Cover and the more general Strip Cover problem, in which each radius may be set finitely-many times. Moreover, we show that the more general class of duty cycle algorithms, in which groups of sensors take turns covering the entire region, can do no better. Finally, we give an optimal O(n^2 log n)-time algorithm for the related Set Radius Strip Cover problem, in which all sensors must be activated immediately.Comment: briefly announced at SPAA 201

Approximating Minimum Independent Dominating Sets in Wireless Networks

We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth. Graphs of bounded growth are used to characterize wireless communication networks, and this class of graph includes many models known from the literature, e.g. (Quasi) Unit Disk Graphs. An independent dominating set is a dominating set in a graph that is also independent. It thus combines the advantages of both structures, and there are many applications that rely on these two structures e.g. in the area of wireless ad hoc networks. The presented approach yields a robust algorithm, that is, the algorithm accepts any undirected graph as input, and returns a (1+")- pproximate minimum dominating set, or a certificate showing that the input graph does not reflect a wireless network

A Trapezoidal Fuzzy Membership Genetic Algorithm (TFMGA) for Energy and Network Lifetime Maximization under Coverage Constrained Problems in Heterogeneous Wireless Sensor Networks

Network lifetime maximization of Wireless Heterogeneous Wireless Sensor Networks (HWSNs) is a difficult problem. Though many methods have been introduced and developed in the recent works to solve network lifetime maximization. However, in HWSNs, the energy efficiency of sensor nodes becomes also a very difficult issue. On the other hand target coverage problem have been also becoming most important and difficult problem. In this paper, new Markov Chain Monte Carlo (MCMC) is introduced which solves the energy efficiency of sensor nodes in HWSN. At initially graph model is modeled to represent HWSNs with each vertex representing the assignment of a sensor nodes in a subset. At the same time, Trapezoidal Fuzzy Membership Genetic Algorithm (TFMGA) is proposed to maximize the number of Disjoint Connected Covers (DCC) and K-Coverage (KC) known as TFMGA-MDCCKC. Based on gene and chromosome information from the TFMGA, the gene seeks an optimal path on the construction graph model that maximizes the MDCCKC. In TFMGA gene thus focuses on finding one more connected covers and avoids creating subsets particularly. A local search procedure is designed to TFMGA thus increases the search efficiency. The proposed TFMGA-MDCCKC approach has been applied to a variety of HWSNs. The results show that the TFMGA-MDCCKC approach is efficient and successful in finding optimal results for maximizing the lifetime of HWSNs. Experimental results show that proposed TFMGA-MDCCKC approach performs better than Bacteria Foraging Optimization (BFO) based approach, Ant Colony Optimization (ACO) method and the performance of the TFMGA-MDCCKC approach is closer to the energy-conserving strategy

Local Approximation Schemes for Ad Hoc and Sensor Networks

We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1+ε)-approximation to the problems at hand for any given ε > 0. The time complexity of both algorithms is O(TMIS + log*! n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every r-neighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs