Showing papers on "Dominating set published in 2006"

Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs

Abstract: For a given graph with weighted vertices, the goal of the minimum-weight dominating set problem is to compute a vertex subset of smallest weight such that each vertex of the graph is contained in the subset or has a neighbor in the subset. A unit disk graph is a graph in which each vertex corresponds to a unit disk in the plane and two vertices are adjacent if and only if their disks have a non-empty intersection. We present the first constant-factor approximation algorithm for the minimum-weight dominating set problem in unit disk graphs, a problem motivated by applications in wireless ad-hoc networks. The algorithm is obtained in two steps: First, the problem is reduced to the problem of covering a set of points located in a small square using a minimum-weight set of unit disks. Then, a constant-factor approximation algorithm for the latter problem is obtained using enumeration and dynamic programming techniques exploiting the geometry of unit disks. Furthermore, we also show how to obtain a constant-factor approximation algorithm for the minimum-weight connected dominating set problem in unit disk graphs.

A simple improved distributed algorithm for minimum CDS in unit disk graphs

Abstract: Several routing schemes in ad hoc networks first establish a virtual backbone and then route messages via backbone nodes. One common way of constructing such a backbone is based on the construction of a connected dominating set (CDS). In this article we present a very simple distributed algorithm for computing a small CDS. Our algorithm has an approximation factor of at most 6.91, improving upon the previous best-known approximation factor of 8 due to Wan et al. [2002]. The improvement relies on a refined analysis of the relationship between the size of a maximal independent set and a minimum CDS in a unit disk graph. This subresult also implies improved approximation factors for many existing algorithm.

On constructing k -connected k -dominating set in wireless ad hoc and sensor networks

Abstract: An important problem in wireless ad hoc and sensor networks is to select a few nodes to form a virtual backbone that supports routing and other tasks such as area monitoring. Previous work in this area has focused on selecting a small virtual backbone for high efficiency. In this paper, we propose the construction of a k-connected k-dominating set (k-CDS) as a backbone to balance efficiency and fault tolerance. Four localized k-CDS construction protocols are proposed. The first protocol randomly selects virtual backbone nodes with a given probability Pk, where Pk depends on the value of k and network condition, such as network size and node density. The second one maintains a fixed backbone node degree of Bk, where Bk also depends on the network condition. The third protocol is a deterministic approach. It extends Wu and Dai's coverage condition, which is originally designed for 1-CDS construction, to ensure the formation of a k-CDS. The last protocol is a hybrid of probabilistic and deterministic approaches. It provides a generic framework that can convert many existing CDS algorithms into k-CDS algorithms. These protocols are evaluated via a simulation study.

Extended Dominating Set and Its Applications in Ad Hoc Networks Using Cooperative Communication

Abstract: We propose a notion of an extended dominating set where each node in an ad hoc network is covered by either a dominating neighbor or several 2-hop dominating neighbors. This work is motivated by cooperative communication in ad hoc networks whereby transmitting independent copies of a packet generates diversity and combats the effects of fading. We first show the NP-completeness of the minimum extended dominating set problem. Then, several heuristic algorithms, global and local, for constructing a small extended dominating set are proposed. These are nontrivial extensions of the existing algorithms for the regular dominating set problem. The application of the extended dominating set in efficient broadcasting is also discussed. The performance analysis includes an analytical study in terms of approximation ratio and a simulation study of the average size of the extended dominating set derived from the proposed algorithms

Optimal transmission radius for energy efficient broadcasting protocols in ad hoc and sensor networks

Abstract: We investigate the problem of minimum energy broadcasting in ad hoc networks where nodes have capability to adjust their transmission range. The minimal transmission energy needed for correct reception by neighbor at distance r is proportional to ralpha+ce , alpha and ce being two environment-dependent constants. We demonstrate the existence of an optimal transmission radius, computed with a hexagonal tiling of the network area, that minimizes the total power consumption for a broadcasting task. This theoretically computed value is experimentally confirmed. The existing localized protocols are inferior to existing centralized protocols for dense networks. We present two localized broadcasting protocols, based on derived "target" radius, that remain competitive for all network densities. The first one, TR-LBOP, computes the minimal radius needed for connectivity and increases it up to the target one after having applied a neighbor elimination scheme on a reduced subset of direct neighbors. In the second one, TRDS, each node first considers only neighbors whose distance is no greater than the target radius (which depends on the power consumption model used), and neighbors in a localized connected topological structure such as RNG or LMST. Then, a connected dominating set is constructed using this subgraph. Nodes not selected for the set may be sent to sleep mode. Nodes in selected dominating set apply TR-LBOP. This protocol is the first one to consider both activity scheduling and minimum energy consumption as one combined problem. Finally, some experimental results for both protocols are given, as well as comparisons with other existing protocols. Our analysis and protocols remain valid if energy needed for packet receptions is charged

Fault-Tolerant Clustering in Ad Hoc and Sensor Networks

Abstract: In this paper, we study distributed approximation algorithms for fault-tolerant clustering in wireless ad hoc and sensor networks. A k-fold dominating set of a graph G = (V,E) is a subset S of V such that every node v \in V \ S has at least k neighbors in S. We study the problem in two network models. In general graphs, for arbitrary parameter t, we propose a distributed algorithm that runs in time O(t^2) and achieves an approximation ratio of O(t\delta^2/t log\delta), where n and \delta denote the number of nodes in the network and the maximal degree, respectively. When the network is modeled as a unit disk graph, we give a probabilistic algorithm that runs in time O(log log n) and achieves an O(1) approximation in expectation. Both algorithms require only small messages of size O(log n) bits.

Energy conservation via domatic partitions

Abstract: Using a dominating set as a coordinator in wireless networks has been proposed in many papers as an energy conservation technique. Since the nodes in a dominating set have the extra burden of coordination, energy resources in such nodes will drain out more quickly than in other nodes. To maximize the lifetime of nodes in the network,it has been proposed that the role of coordinators be rotated among the nodes in the network. One abstraction that has been considered for the problem of picking a collection of coordinators and cycling through them, is the domatic partition problem. This is the problem of partitioning the set of the nodes of the network into dominating sets with the aim of maximizing the number of dominating sets. In this paper,we consider the k -domatic partition problem. A k -dominating set is a subset D of nodes such that every node in the network is at distance at most k from D. The k-domatic partition problem seeks to partition the network into maximum number of k-dominating sets.We point out that from the point of view of saving energy,it may be better to construct a k-domatic partition for k >1.We present three deterministic, distributed algorithms for finding large k-domatic partitions for k > 1. Each of our algorithms constructs a k-domatic partition of size at least a constant fraction of the largest possible (k 1)-domatic partition. Our first algorithm runs in constant time on unit ball graphs (UBGs) in Euclidean space assuming that all nodes know their positions in a global coordinate system. Our second algorithm drops knowledge of global coordinates and instead assumes that pairwise distances between neighboring nodes are known. This algorithm runs in O(log* n ) time on UBGs in a metric space with constant doubling dimension. Our third algorithm drops all reliance on geometric information, using connectivity information only. This algorithm runs in O(log Δ · log *n) time on growth-bounded graphs. Euclidean UBGs, UBGs in metric spaces with constant doubling dimension, and growth-bounded graphs are successively more general models of wireless networks and all three models include the well-known, but somewhat simplistic wireless network models such as unit disk graphs.

Parameterized complexity of independence and domination on geometric graphs

Abstract: We investigate the parameterized complexity of MAXIMUM INDEPENDENT SET and DOMINATING SET restricted to certain geometric graphs. We show that DOMINATING SET is W[1]-hard for the intersection graphs of unit squares, unit disks, and line segments. For MAXIMUM INDEPENDENT SET, we show that the problem is W[1]-complete for unit segments, but fixed-parameter tractable if the segments are axis-parallel.

Edge dominating set : Efficient enumeration-based exact algorithms

Abstract: We analyze EDGE DOMINATING SET from a parameterized perspective. More specifically, we prove that this problem is in FPT for general (weighted) graphs. The corresponding algorithms rely on enumeration techniques. In particular, we show how the use of compact representations may speed up the decision algorithm.

Dynamic programming and fast matrix multiplication

Abstract: We give a novel general approach for solving NP-hard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We show that our approach works faster than the usual dynamic programming solution for any vertex subset problem on graphs of bounded branchwidth. In particular, we obtain the fastest algorithms for Planar Independent Set of runtime O(22.52√n), for PLANAR DOMINATING SET of runtime exact O(23.99√n) and parameterized O(211.98√k)ċ nO(1), and for PLANAR HAMILTONIAN Cycle of runtime O(25.58√n). The exponent of the running time is depending heavily on the running time of the fastest matrix multiplication algorithm that is currently o(n2.376).

Solving connected dominating set faster than 2n

Abstract: In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves. Despite its relevance in applications, the best known exact algorithm for the problem is the trivial Ω(2 n ) algorithm which enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such difference is not surprising, since connectivity is a global property, and non-local problems are typically much harder to solve exactly. In this paper we break the 2 barrier, by presenting a simple O(1.9407 n ) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique.

Approximation hardness of edge dominating set problems

Abstract: We provide the first interesting explicit lower bounds on efficient approximability for two closely related optimization problems in graphs, MINIMUM EDGE DOMINATING SET and MINIMUM MAXIMAL MATCHING. We show that it is NP-hard to approximate the solution of both problems to within any constant factor smaller than $${\frac{7}{6}}$$ . The result extends with negligible loss to bounded degree graphs and to everywhere dense graphs.

Nonblocker: parameterized algorithmics for minimum dominating set

Abstract: We provide parameterized algorithms for nonblocker, the parametric dual of the well known dominating set problem. We exemplify three methodologies for deriving parameterized algorithms that can be used in other circumstances as well, including the (i) use of extremal combinatorics (known results from graph theory) in order to obtain very small kernels, (ii) use of known exact algorithms for the (nonparameterized) minimum dominating set problem, and (iii) use of exponential space. Parameterized by the size kd of the non-blocking set, we obtain an algorithm that runs in time ${\mathcal O}^{*}(1.4123^{k_{d}})$ when allowing exponential space.

EDGE DOMINATING SET: efficient enumeration-based exact algorithms

Abstract: We analyze edge dominating set from a parameterized perspective. More specifically, we prove that this problem is in ${\mathcal{FPT}}$ for general (weighted) graphs. The corresponding algorithms rely on enumeration techniques. In particular, we show how the use of compact representations may speed up the decision algorithm.

A self-stabilizing minimal dominating set algorithm with safe convergence

Abstract: A self-stabilizing distributed system is a fault-tolerant distributed system that tolerates any kind and any finite number of transient faults, such as message loss and memory corruption. In this paper, we formulate a concept of safe convergence in the framework of self-stabilization. An ordinary self-stabilizing algorithm has no safety guarantee while it is in converging from any initial configuration. The safe convergence property guarantees that a system quickly converges to a safe configuration, and then, it gracefully moves to an optimal configuration without breaking safety. Then, we propose a minimal independent dominating set algorithm with safe convergence property. Especially, the proposed algorithm computes the lexicographically first minimal independent dominating set according to the process identifier as a priority. The priority scheme can be arbitrarily changed such as stability, battery power and/or computation power of node.

Graphs with large paired-domination number

Abstract: In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (1998) Networks 32: 199–206. A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by $$\gamma_{\rm pr}(G)$$ , is the minimum cardinality of a paired-dominating set of G. Let G be a connected graph of order n with minimum degree at least two. Haynes and Slater (1998) Networks 32: 199–206, showed that if n ≥ 6, then $$\gamma_{\rm pr}(G) \le 2n/3$$ . In this paper, we show that there are exactly ten graphs that achieve equality in this bound. For n ≥ 14, we show that $$\gamma_{\rm pr}(G) \le 2(n-1)/3$$ and we characterize the (infinite family of) graphs that achieve equality in this bound.

Strongly connected dominating sets in wireless sensor networks with unidirectional links

Abstract: A Connected Dominating Set (CDS) can serve as a virtual backbone for a wireless sensor network since there is no fixed infrastructure or centralized management in wireless sensor networks. With the help of CDS, routing is easier and can adapt quickly to network topology changes. The CDS problem has been studied extensively in undirected graphs, especially in unit disk graphs, in which each sensor node has the same transmission range. However, in practice, the transmission ranges of all nodes are not necessarily to be equal. In this paper, we model a network as a disk graph where unidirectional links are considered and introduce the Strongly Connected Dominating Set (SCDS) problem in disk graphs. We propose two constant approximation algorithms for the SCDS problem and compare their performances through the theoretical analysis.

Solving connected dominating set faster than 2 n

Abstract: In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves. Despite its relevance in applications, the best known exact algorithm for the problem is the trivial Ω(2n) algorithm which enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such difference is not surprising, since connectivity is a global property, and non-local problems are typically much harder to solve exactly. In this paper we break the 2n barrier, by presenting a simple O(1.9407n) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique.

On the Independent Domination Number of Random Regular Graphs

Abstract: A dominating set $\cal D$ of a graph $G$ is a subset of $V(G)$ such that, for every vertex $v\in V(G)$, either in $v\in {\cal D}$ or there exists a vertex $u \in {\cal D}$ that is adjacent to $v$. We are interested in finding dominating sets of small cardinality. A dominating set $\cal I$ of a graph $G$ is said to be independent if no two vertices of ${\cal I}$ are connected by an edge of $G$. The size of a smallest independent dominating set of a graph $G$ is the independent domination number of $G$. In this paper we present upper bounds on the independent domination number of random regular graphs. This is achieved by analysing the performance of a randomized greedy algorithm on random regular graphs using differential equations.

NONBLOCKER : Parameterized algorithmics for MINIMUM DOMINATING SET

Abstract: We provide parameterized algorithms for NONBLOCKER, the parametric dual of the well known DOMINATING SET problem. We exemplify three methodologies for deriving parameterized algorithms that can be used in other circumstances as well, including the (i) use of extremal combinatorics (known results from graph theory) in order to obtain very small kernels, (ii) use of known exact algorithms for the (nonparameterized) MINIMUM DOMINATING SET problem, and (iii) use of exponential space. Parameterized by the size κ d of the non-blocking set, we obtain an algorithm that runs in time O*(1.4123 k d) when allowing exponential space.