Showing papers on "Dominating set published in 2005"

Bidimensionality: new connections between FPT algorithms and PTASs

Abstract: We demonstrate a new connection between fixed-parameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of so-called "bidimensional" problems to show that essentially all such problems have both subexponential fixed-parameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal problems, dominating set, edge dominating set, r-dominating set, diameter, connected dominating set, connected edge dominating set, and connected r-dominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two well-known problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the Lipton-Tarjan separator approach [FOCS'77] and the Baker layerwise decomposition approach [FOCS'83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a "nonlocal" structure.

Constant-time distributed dominating set approximation

Abstract: Finding a small dominating set is one of the most fundamental problems of classical graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary, possibly constant parameter k and maximum node degree Δ, our algorithm computes a dominating set of expected size O(kΔ2/k log (Δ)|DSOPT|) in O (K2) rounds. Each node has to send O(k2 Δ) messages of size O(log Δ). This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds.

On greedy construction of connected dominating sets in wireless networks

Abstract: Summary Since no fixed infrastructure and no centralized management present in wireless networks, a connected dominating set (CDS) of the graph representing the network is widely used as a virtual backbone. Constructing a minimum CDS is NP-hard. In this paper, we propose a new greedy algorithm, called S-MIS, with the help of Steiner tree that can construct a CDS within a factor of 4:8 þ ln5 from the optimal solution. We also introduce the distributed version of this algorithm. We prove that the proposed algorithm is better than the current best performance ratio which is 6.8. A simulation is conducted to compare S-MIS with its variation which is rS-MIS. The simulation shows that the sizes of the CDSs generated by S-MIS and rS-MIS are almost the same. Copyright # 2005 John Wiley & Sons, Ltd.

On constructing k-connected k-dominating set in wireless networks

Abstract: An important problem in wireless networks, such as 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. We propose to construct a k-connected k-dominating set (k-CDS) as a backbone to balance efficiency and fault tolerance. Three localized k-CDS construction protocols are proposed. The first protocol randomly selects virtual backbone nodes with a given probability p/sub k/, where p/sub k/ depends on network condition and the value of k. The second 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.

Tight lower bounds for certain parameterized NP-hard problems

Abstract: Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time n^o^(^k^)m^O^(^1^), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t-1)-st level W[t-1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat, hitting set, set cover, and feature set, cannot be solved in time n^o^(^k^)m^O^(^1^), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W[1] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weightedq-sat (for any fixed q>=2), clique, independent set, and dominating set, cannot be solved in time n^o^(^k^) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n^km^O^(^1^) or O(n^k).

Measure and conquer: domination – a case study

Abstract: Davis-Putnam-style exponential-time backtracking algorithms are the most common algorithms used for finding exact solutions of NP-hard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the subproblems is defined; this measure is used to lower bound the progress made by the algorithm at each branching step. For the last 30 years the research on exact algorithms has been mainly focused on the design of more and more sophisticated algorithms. However, measures used in the analysis of backtracking algorithms are usually very simple. In this paper we stress that a more careful choice of the measure can lead to significantly better worst case time analysis. As an example, we consider the minimum dominating set problem. The currently fastest algorithm for this problem has running time O(20.850n) on n-nodes graphs. By measuring the progress of the (same) algorithm in a different way, we refine the time bound to O(20.598 n). A good choice of the measure can provide such a (surprisingly big) improvement; this suggests that the running time of many other exponential-time recursive algorithms is largely overestimated because of a “bad” choice of the measure.

Fast deterministic distributed maximal independent set computation on growth-bounded graphs

Abstract: The distributed complexity of computing a maximal independent set in a graph is of both practical and theoretical importance While there exists an elegant O(log n) time randomized algorithm for general graphs [20], no deterministic polylogarithmic algorithm is known In this paper, we study the problem in graphs with bounded growth, an important family of graphs which includes the well-known unit disk graph and many variants thereof Particularly, we propose a deterministic algorithm that computes a maximal independent set in time O(log Δ· log*n) in graphs with bounded growth, where n and Δ denote the number of nodes and the maximal degree in G, respectively

Maximizing the lifetime of dominating sets

Abstract: We investigate the problem of maximizing the lifetime of wireless ad hoc and sensor networks. Being battery powered, nodes in such networks have to perform their intended task under rigid energy restrictions that forces the designers to impose a judicious power management and scheduling. For the purpose of saving energy, dominating set based clustering has turned out to be a useful and generic concept in such networks. In data gathering applications, for example, only nodes in the dominating set must be active, while all other nodes can remain in the energy-efficient sleep mode. Prolonging the duration of such a dominating set based clustering is a key algorithmic challenge. In this paper, we define the maximum cluster-lifetime problem which asks for a schedule that maximizes the time the network is clustered by a dominating set. We give approximation algorithms with an approximation ratio of O(log n) for several variants of the maximum cluster-lifetime problem. Our approach is based on results given in a paper by Feige, Ilalldorsson, Kortsarz, and Srinivasan on the domatic partition problem.

On double domination in graphs

Abstract: In a graph G, a vertex dominates itself and its neighbors. A subset S µ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number ∞£2(G). A function f(p) is deflned, and it is shown that ∞£2(G) = minf(p), where the minimum is taken over the n-dimensional cube C n = fp = (p1;:::;pn) j pi 2 IR;0 • pi • 1;i = 1;:::;ng. Using this result, it is then shown that if G has order n with minimum degree ‐ and average degree d, then ∞£2(G) • ((ln(1 + d) + ln‐ + 1)=‐)n.

A PTAS for the minimum dominating set problem in unit disk graphs

Abstract: We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input. The runtime of the PTAS is nO(1/elog 1/e). The algorithm accepts any undirected graph as input, and returns a (1+e)-approximate minimum dominating set, or a certificate showing that the input graph is no unit disk graph, making the algorithm robust. The PTAS can easily be adapted to other classes of geometric intersection graphs.

Local approximation schemes for ad hoc and sensor networks

Abstract: 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+e)-approximation to the problems at hand for any given e > 0. The time complexity of both algorithms is O(TMIS + log*! n/eO(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.

Preserving area coverage in wireless sensor networks by using surface coverage relay dominating sets

Abstract: Sensor networks consist of autonomous nodes with limited battery and of base stations with theoretical infinite energy. Nodes can be sleep to extend the lifespan of the network without compromising neither area coverage nor network connectivity. This paper addresses the area coverage problem with equal sensing and communicating radii. The goal is to minimize the number of active sensors involved in coverage task, while computing a connected set able to report to monitoring stations. Our solution is fully localized, and each sensor is able to make decision on whether to sleep or to be active based on two messages sent by each sensor. The first message is a "hello" message to gather position of all neighboring nodes. Then each node computes its own relay area dominating set, by taking the furthest neighbor as the first node, and then adding neighbors farthest to the isobarycenter of already selected neighbors, until the area covered by neighbors is fully covered. The second message broadcasts this relay set to neighbors. Each node decides to be active if it has highest priority among its neighbors or is a relay node for its neighbor with the highest priority.

Parametric duality and kernelization: lower bounds and upper bounds on kernel size

Abstract: We develop new techniques to derive lower bounds on the kernel size for certain parameterized problems. For example, we show that unless $\mathcal{P}$=$\mathcal{NP}$, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. We derive an upper bound of 67k on the problem kernel for planar dominating set improving the previous 335k upper bound by Alber et al.

Locating Stops Along Bus or Railway Lines—A Bicriteria Problem

Abstract: In this paper we consider the location of stops along the edges of an already existing public transportation network. This can be the introduction of bus stops along given routes, or of railway stations along the tracks in a railway network. The goal is to achieve a maximal covering of given demand points with a minimal number of stops. This bicriteria problem is in general NP-hard. We present a finite dominating set yielding an IP-formulation as a bicriteria set covering problem. Using this formulation we discuss cases in which the bicriteria stop location problem can be solved in polynomial time. Extensions for tackling real-world instances are mentioned.

Improved algorithms and complexity results for power domination in graphs

Abstract: The POWER DOMINATING SET problem is a variant of the classical domination problem in graphs: Given an undirected graph G = (V, E), find a minimum P C V such that all vertices in V are observed by vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that POWER DOMINATING SET can be solved by bounded-treewidth dynamic programs. Moreover, we simplify and extend several NP-completeness results, particularly showing that POWER DOMINATING SET remains NP-complete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that POWER DOMINATING SET parameterized by |P| is W[2]-hard and cannot be better approximated than DOMINATING SET.

Solving the k-center Problem Efficiently with a Dominating Set Algorithm

Abstract: We present a polynomial time heuristic algorithm for the minimum dominating set problem. The algorithm can readily be used for solving the minimum alpha-all-neighbor dominating set problem and the minimum set cover problem. We apply the algorithm in heuristic solving the minimum k-center problem in polynomial time. Using a standard set of 40 test problems we experimentally show that our k-center algorithm performs much better than other well-known heuristics and is competitive with the best known (non-polynomial time) algorithms for solving the k-center problem in terms of average quality and deviation of the results as well as execution time.

On total restrained domination in graphs

Abstract: In this paper we initiate the study of total restrained domination in graphs. Let G = (V,E) be a graph. A total restrained dominating set is a set S $$ \subseteq $$ V where every vertex in V - S is adjacent to a vertex in S as well as to another vertex in V - S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ (G), is the smallest cardinality of a total restrained dominating set of G. First, some exact values and sharp bounds for γ (G) are given in Section 2. Then the Nordhaus-Gaddum-type results for total restrained domination number are established in Section 3. Finally, we show that the decision problem for γ (G) is NP-complete even for bipartite and chordal graphs in Section 4.

Lower Bounds and Algorithms for Dominating Sets in Web Graphs

Abstract: In this paper we study the size of generalised dominating sets in two graph processes that are widely used to model aspects of the World Wide Web. On the one hand, we show that graphs generated this way have fairly large dominating sets (i.e., linear in the size of the graph). On the other hand, we present efficient strategies to construct small dominating sets. The algorithmic results represent an application of a particular analysis technique which can be used to characterise the asymptotic behaviour of a number of dynamic processes related to the web.

Approximation algorithms for unit disk graphs

Abstract: We consider several graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set) relevant to mobile ad hoc networks. We propose two new notions: thickness and density. If the thickness of a unit disk graph is bounded, then the mentioned problems can be solved in polynomial time. For unit disk graphs of bounded density, we present a new asymptotic fully-polynomial approximation scheme for the considered problems. The scheme for Minimum Connected Dominating Set is the first Baker-like asymptotic FPTAS for this problem. By adapting the proof, it implies e.g. an asymptotic FPTAS for Minimum Connected Dominating Set on planar graphs.

Bounding the number of minimal dominating sets: a measure and conquer approach

Abstract: We show that the number of minimal dominating sets in a graph on n vertices is at most 1.7697n, thus improving on the trivial $\mathcal{O}(2^{n}/\sqrt{n})$ bound. Our result makes use of the measure and conquer technique from exact algorithms, and can be easily turned into an $\mathcal{O}(1.7697^{n})$ listing algorithm. Based on this result, we derive an $\mathcal{O}(2.8805^{n})$ algorithm for the domatic number problem, and an $\mathcal{O}(1.5780^{n})$ algorithm for the minimum-weight dominating set problem. Both algorithms improve over the previous algorithms.

Improved algorithms and complexity results for power domination in graphs

Abstract: The Power Dominating Set problem is a variant of the classical domination problem in graphs: Given an undirected graph G=(V,E), find a minimum P⊆V such that all vertices in V are “observed” by vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that Power Dominating Set can be solved by “bounded-treewidth dynamic programs.” Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that Power Dominating Set parameterized by |P| is W[2]-hard and cannot be better approximated than Dominating Set.

A timer-based protocol for connected dominating set construction in IEEE 802.11 multihop mobile ad hoc networks

Abstract: Connected dominating set has been used widely in multihop ad hoc networks (MANET) by numerous routing, broadcast and collision avoidance protocols Although computing minimum connected dominating set is known to be NP-hard, many protocols have been proposed to construct a suboptimal dominating set However, these protocols are either too complicated, needing nonlocal information, or not adaptive to topology changes In this paper, we present a MAC-layer timer-based connected dominating set construction protocol In our protocol, candidate nodes set up a timer based on the number of uncovered neighbors and determine whether or not to join the dominating set when the timer expires The protocol is simple, distributed, inexpensive, and adaptive to station mobility The simulation results show that our protocol can construct connected dominating set using 35% to 60% less nodes than other distributed connected dominating set protocols

Generating quality dominating sets for sensor network

Abstract: We consider the problem of generating dominating sets for applications in information communication and sensor network. Known algorithms for solving this problem consider number of nodes in the dominating set as the sole criteria. We report algorithms for generating dominating sets by considering diameter and interference as the additional factors. Experimental investigation shows that the proposed algorithms can generate diameter reduced and interference aware dominating sets without increasing the size of the solution.

Maintaining weakly-connected dominating sets for clustering ad hoc networks

Abstract: An ad hoc network is a multihop wireless communication network supporting mobile users. Network performance degradation is a major problem as the network becomes larger. Clustering is an approach to simplify the network structure and thus alleviate the scalability problem. One method that has been proposed to form clusters is to use weakly-connected dominating sets [Y.P. Chen, A.L. Liestman, Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks, in: The Third ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc'02), 2002, pp. 165-172; Y.P. Chen, A.L. Liestman, A zonal algorithm for clustering ad hoc networks, International Journal of Foundations of Computer Science 14(2) (2003) 305-322]. Here, we present a zonal distributed algorithm to maintain weakly-connected dominating sets as the network structure changes. When the zones are small, the algorithm is essentially localized; when the zones are large, it behaves more globally. The size of the weakly-connected dominating set obtained also varies depending on the choice of zone size, with larger zones generally resulting in smaller weakly-connected dominating sets. Experiments provide evidence that this maintenance algorithm keeps the size of the weakly-connected dominating set approximately the same as its initial size and does not compromise the network connectivity.

Domination in the families of Frank and Hamacher t-norms

Abstract: Domination is a relation between general operations defined on a poset. The old open problem is whether domination is transitive on the set of all t-norms. In this paper we contribute partially by inspection of domination in the family of Frank and Hamacher t-norms. We show that between two different t-norms from the same family, the domination occurs iff at least one of the t-norms involved is a maximal or minimal member of the family. The immediate consequence of this observation is the transitivity of domination on both inspected families of t-norms.

A new distributed approximation algorithm for constructing minimum connected dominating set in wireless ad hoc networks

Abstract: In recent years, constructing a virtual backbone by nodes in a connected dominating set (CDS) has been proposed to improve the performance of ad hoc wireless networks. In general, a dominating set satisfies that every vertex in the graph is either in the set or adjacent to a vertex in the set. A CDS is a dominating set that also induces a connected sub-graph. However, finding the minimum connected dominating set (MCDS) is a well-known NP-hard problem in graph theory. Approximation algorithms for MCDS have been proposed in the literature. Most of these algorithms suffer from a poor approximation ratio, and from high time complexity and message complexity. In this paper, we present a new distributed approximation algorithm that constructs a MCDS for wireless ad hoc networks based on a maximal independent set (MIS). Our algorithm, which is fully localized, has a constant approximation ratio, and O(n) time and O(n) message complexity. In this algorithm, each node only requires the knowledge of its one-hop neighbours and there is only one shortest path connecting two dominators that are at most three hops away. We not only give theoretical performance analysis for our algorithm, but also conduct extensive simulation to compare our algorithm with other algorithms in the literature. Simulation results and theoretical analysis show that our algorithm has better efficiency and performance than others. Copyright © 2005 John Wiley & Sons, Ltd.