Showing papers on "Dominating set published in 2009"

A measure & conquer approach for the analysis of exact algorithms

Abstract: For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call “Measure & Conquer”, as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis.In order to show the potentialities of Measure & Conquer, we consider two well-studied NP-hard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis).Our examples show that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.

Meta) Kernelization

Abstract: Polynomial time preprocessing to reduce instance size is one of the most commonly deployed heuristics to tackle computationally hard problems. In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, we can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this paper, we show that all problems expressible in Counting Monadic Second Order Logic and satisfying a compactness property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker compactness condition admit a linear kernel on graphs of bounded genus. The study of kernels on planar graphs was initiated by a seminal paper of Alber, Fellows, and Niedermeier [J. ACM, 2004 ] who showed that Planar Dominating Set admits a linear kernel. Following this result, a multitude of problems have been shown to admit linear kernels on planar graphs by combining the ideas of Alber et al. with problem specific reduction rules. Our theorems unify and extend all previously known kernelization results for planar graph problems. Combining our theorems with the Erdos-Posa property we obtain various new results on linear kernels for a number of packing and covering problems.

Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs

Abstract: There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a k O(dk) n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain K h as a topological minor, we give an improved algorithm for the problem with running time (O(h)) hk n. For graphs which are K h -minor-free, the running time is further reduced to (O(log h)) hk/2 n. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an H-minor-free graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(nlog n) worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs.

A better constant-factor approximation for weighted dominating set in unit disk graph

Abstract: This paper presents a (10+e)-approximation algorithm to compute minimum-weight connected dominating set (MWCDS) in unit disk graph. MWCDS is to select a vertex subset with minimum weight for a given unit disk graph, such that each vertex of the graph is contained in this subset or has a neighbor in this subset. Besides, the subgraph induced by this vertex subset is connected. Our algorithm is composed of two phases: the first phase computes a dominating set, which has approximation ratio 6+e (e is an arbitrary positive number), while the second phase connects the dominating sets computed in the first phase, which has approximation ratio 4.

Positive Influence Dominating Set in Online Social Networks

Abstract: Online social network has developed significantly in recent years as a medium of communicating, sharing and disseminating information and spreading influence. Most of current research has been on understanding the property of online social network and utilizing it to spread information and ideas. In this paper, we explored the problem of how to utilize online social networks to help alleviate social problems in the physical world, for example, the drinking, smoking, and drug related problems. We proposed a Positive Influence Dominating Set (PIDS) selection algorithm and analyzed its effect on a real online social network data set through simulations. By comparing the size and the average positive degree of PIDS with those of a 1-dominating set, we found that by strategically choosing 26% more people into the PIDS to participate in the intervention program, the average positive degree increases by approximately 3.3 times. In terms of the application, this result implies that by moderately increasing the participation related cost, the probability of positive influencing the whole community through the intervention program is significantly higher. We also discovered that a power law graph has empirically larger dominating sets (both the PIDS and 1-dominating set) than a random graph does.

Domination Problems in Nowhere-Dense Classes

Abstract: We investigate the parameterized complexity of generalisations and variations of the dominating set problem on classes of graphs that are nowhere dense. In particular, we show that the distance-$d$ dominating-set problem, also known as the $(k,d)$-centres problem, is fixed-parameter tractable on any class that is nowhere dense and closed under induced subgraphs. This generalises known results about the dominating set problem on $H$-minor free classes, classes with locally excluded minors and classes of graphs of bounded expansion. A key feature of our proof is that it is based simply on the fact that these graph classes are uniformly quasi-wide, and does not rely on a structural decomposition. Our result also establishes that the distance-$d$ dominating-set problem is FPT on classes of bounded expansion, answering a question of Ne{\v s}et{\v r}il and Ossona de Mendez.

Approximation Algorithms and Hardness for Domination with Propagation

Abstract: The Power Dominating Set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes $S$ that power dominates all the nodes, where a node $v$ is power dominated if (1) $v$ is in $S$ or $v$ has a neighbor in $S$, or (2) $v$ has a neighbor $w$ such that $w$ and all of its neighbors except $v$ are power dominated. We show a hardness of approximation threshold of $2^{\log^{1-\epsilon}n}$ in contrast to the logarithmic hardness for the dominating set problem. We give an $O(\sqrt{n})$-approximation algorithm for planar graphs and show that our methods cannot improve on this approximation guarantee. Finally, we initiate the study of PDS on directed graphs and show the same hardness threshold of $2^{\log^{1-\epsilon}n}$ for directed acyclic graphs. Also we show that the directed PDS problem can be solved optimally in linear time if the underlying undirected graph has bounded tree-width.

Analysis on theoretical bounds for approximating dominating set problems

Abstract: Connected Dominating Set is widely used as virtual backbone in wireless networks to improve network performance and optimize routing protocols. Based on special characteristics of ad-hoc and sensor networks, we usually use unit disk graph to represent the corresponding geometrical structures, where each node has a unit transmission range and two nodes are said to be adjacent if the distance between them is less than 1. Since every Maximal Independent Set (MIS) is a dominating set and it is easy to construct, we can firstly find an MIS and then connect it into a Connected Dominating Set (CDS). Therefore, the ratio to compare the size of an MIS with a minimum CDS becomes a theoretical upper bound for approximation algorithms to compute CDS. In our paper, with the help of Voronoi diagram and Euler's formula, we improved this upper bound, so that improved the approximations based on this relation.

A Power Aware Minimum Connected Dominating Set for Wireless Sensor Networks

Abstract: Connected Dominating Set (CDS) problem in unit disk graph has a significant impact on an efficient design of routing protocols in wireless sensor networks. In this paper, an algorithm is proposed for finding Minimum Connected Dominating Set (MCDS) using Dominating Set. Dominating Sets are connected by using Steiner tree. The algorithm goes through three phases. In first phase Dominating Sets are found, in second phase connectors are identified, connected through Steiner tree. In third phase the CDS obtained in second phase is pruned to obtain a MCDS. MCDS so constructed needs to adapt to the continual topology changes due to deactivation of a node due to exhaustion of battery power. These topological changes due to power constraints are taken care by a local repair algorithm that reconstructs the MCDS i.e. Power Aware MCDS, using only neighbourhood information. Simulation results indicate both the heuristics are very efficient and result in near optimal MCDS.

A Neural Network Model to Minimize the Connected Dominating Set for Self-Configuration of Wireless Sensor Networks

Abstract: A wireless ad hoc sensor network consists of a number of sensors spreading across a geographical area. The performance of the network suffers as the number of nodes grows, and a large sensor network quickly becomes difficult to manage. Thus, it is essential that the network be able to self-organize. Clustering is an efficient approach to simplify the network structure and to alleviate the scalability problem. One method to create clusters is to use weakly connected dominating sets (WCDSs). Finding the minimum WCDS in an arbitrary graph is an NP-complete problem. We propose a neural network model to find the minimum WCDS in a wireless sensor network. We present a directed convergence algorithm. The new algorithm outperforms the normal convergence algorithm both in efficiency and in the quality of solutions. Moreover, it is shown that the neural network is robust. We investigate the scalability of the neural network model by testing it on a range of sized graphs and on a range of transmission radii. Compared with Guha and Khuller's centralized algorithm, the proposed neural network with directed convergency achieves better results when the transmission radius is short, and equal performance when the transmission radius becomes larger. The parallel version of the neural network model takes time O(d) , where d is the maximal degree in the graph corresponding to the sensor network, while the centralized algorithm takes O(n 2). We also investigate the effect of the transmission radius on the size of WCDS. The results show that it is important to select a suitable transmission radius to make the network stable and to extend the lifespan of the network. The proposed model can be used on sink nodes in sensor networks, so that a sink node can inform the nodes to be a coordinator (clusterhead) in the WCDS obtained by the algorithm. Thus, the message overhead is O(M), where M is the size of the WCDS.

A (4+ε)-approximation for the minimum-weight dominating set problem in unit disk graphs

Abstract: We present a (4+e)-approximation algorithm for the problem of computing a minimum-weight dominating set in unit disk graphs, where e is an arbitrarily small constant. The previous best known approximation ratio was 5+e. The main result of this paper is a 4-approximation algorithm for the problem restricted to constant-size areas. To obtain the (4+e)-approximation algorithm for the unrestricted problem, we then follow the general framework from previous constant-factor approximations for the problem: We consider the problem in constant-size areas, and combine the solutions obtained by our 4-approximation algorithm for the restricted case to get a feasible solution for the whole problem. Using the shifting technique (selecting a best solution from several considered partitionings of the problem into constant-size areas) we obtain the claimed (4+e)-approximation algorithm. By combining our algorithm with a known algorithm for node-weighted Steiner trees, we obtain a 7.875-approximation for the minimum-weight connected dominating set problem in unit disk graphs.

A PTAS for minimum connected dominating set in 3-dimensional Wireless sensor networks

Abstract: When homogeneous sensors are deployed into a three-dimensional space instead of a plane, the mathematical model for the sensor network is a unit ball graph instead of a unit disk graph. It is known that for the minimum connected dominating set in unit disk graph, there is a polynomial time approximation scheme (PTAS). However, that construction cannot be extended to obtain a PTAS for unit ball graph. In this paper, we will introduce a new construction, which gives not only a PTAS for the minimum connected dominating set in unit ball graph, but also improves running time of PTAS for unit disk graph.

Tighter Approximation Bounds for Minimum CDS in Wireless Ad Hoc Networks

Abstract: Connected dominating set (CDS) has a wide range of applications in wireless ad hoc networks A number of approximation algorithms for constructing a small CDS in wireless ad hoc networks have been proposed in the literature The majority of these algorithms follow a general two-phased approach The first phase constructs a dominating set, and the second phase selects additional nodes to interconnect the nodes in the dominating set In the performance analyses of these two-phased algorithms, the relation between the independence number ? and the connected domination number ? c of a unit-disk graph plays the key role The best-known relation between them is $\alpha\leq3\frac{2}{3}\gamma_{c}+1$ In this paper, we prove that ? ≤ 34306? c + 48185 This relation leads to tighter upper bounds on the approximation ratios of two approximation algorithms proposed in the literature

Solving Dominating Set in Larger Classes of Graphs: FPT Algorithms and Polynomial Kernels

Abstract: We show that the k -Dominating Set problem is fixed parameter tractable (FPT) and has a polynomial kernel for any class of graphs that exclude K i,j as a subgraph, for any fixed i,j ≥ 1. This strictly includes every class of graphs for which this problem has been previously shown to have FPT algorithms and/or polynomial kernels. In particular, our result implies that the problem restricted to bounded-degenerate graphs has a polynomial kernel, solving an open problem posed by Alon and Gutner in [3].

Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction

Abstract: We explore three important avenues of research in algorithmic graph-minor theory, which all stem from a key min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner’s Conjecture about the structure of minor-closed graph properties. First, we obtain the only known polynomial min-max relation for graphs that do not exclude any fixed minor, namely, map graphs and power graphs. Second, we obtain explicit (and improved) bounds on the min-max relation for an important class of graphs excluding a minor, namely, K 3,k -minor-free graphs, using new techniques that do not rely on Graph Minor Theory. These two avenues lead to faster fixed-parameter algorithms for two families of graph problems, called minor-bidimensional and contraction-bidimensional parameters, which include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set, connected R-dominating set, and unweighted TSP tour. Third, we disprove a variation of Wagner’s Conjecture for the case of graph contractions in general graphs, and in a sense characterize which graphs satisfy the variation. This result demonstrates the limitations of a general theory of algorithms for the family of contraction-closed problems (which includes, for example, the celebrated dominating-set problem). If this conjecture had been true, we would have had an extremely powerful tool for proving the existence of efficient algorithms for any contraction-closed problem, like we do for minor-closed problems via Graph Minor Theory.

Dominating sets and Domination polynomials of Cycles

Abstract: Let G=(V,E) be a simple graph. A set S\subset V is a dominating set of G, if every vertex in V\S is adjacent to at least one vertex in S. Let {\mathcal C}_n^i be the family of dominating sets of a cycle C_n with cardinality i, and let d(C_n,i) = |{\mathcal C}_n^i. In this paper, we construct {\mathcal C}_n^i, and obtain a recursive formula for d(C_n, i). Using this recursive formula, we consider the polynomial D(C_n, x) = \sum_{i=1}^n d(C_n, i)x^i, which we call domination polynomial of cycles and obtain some properties of this polynomial.

Inclusion/Exclusion Meets Measure and Conquer Exact Algorithms for Counting Dominating Sets

Abstract: In this paper, two central techniques from the field of ex- ponential time algorithms are combined for the first time: inclusion/ex- clusion and branching with measure and conquer analysis. In this way, we have obtained an algorithm that, for each κ ,c ounts the number of dominating sets of size κ in O(1.5048 n ) time. This algorithm improves the previously fastest algorithm that counts the number of minimum dominating sets. The algorithm is even slightly faster than the previous fastest algorithm for minimum dominating set, thus improving this result while computing much more information. When restricted to c-dense graphs, circle graphs, 4-chordal graphs or weakly chordal graphs, our combination of branching with inclusion/ex- clusion leads to significantly faster counting and decision algorithms than the previously fastest algorithms for dominating set. All results can be extended to counting (minimum) weight dominating sets when the size of the set of possible weight sums is polynomially bounded.

Solving Dominating Set in Larger Classes of Graphs: FPT Algorithms and Polynomial Kernels

Abstract: We show that the k-Dominating Set problem is fixed parameter tractable (FPT) and has a polynomial kernel for any class of graphs that exclude K_{i,j} as a subgraph, for any fixed i, j >= 1. This strictly includes every class of graphs for which this problem has been previously shown to have FPT algorithms and/or polynomial kernels. In particular, our result implies that the problem restricted to bounded- degenerate graphs has a polynomial kernel, solving an open problem posed by Alon and Gutner.

Subexponential Algorithms for Partial Cover Problems

Abstract: Partial Cover problems are optimization versions of fundamental and well studied problems like {\sc Vertex Cover} and {\sc Dominating Set}. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number ($k$) of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by $k$. It was recently shown by Amini et. al. [{\em FSTTCS 08}\,] that {\sc Partial Vertex Cover} and {\sc Partial Dominating Set} are fixed parameter tractable on large classes of sparse graphs, namely $H$-minor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time $2^{\cO(k)}n^{\cO(1)}$.

Planar Capacitated Dominating Set Is W[1]-Hard

Abstract: Given a graph G together with a capacity function c : V(G) ??, we call S ? V(G) a capacitated dominating set if there exists a mapping f: (V(G) ? S) ?S which maps every vertex in (V(G) ? S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ? S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a capacity function c and a positive integer k and asked whether G has a capacitated dominating set of size at most k. In this paper we show that Planar Capacitated Dominating Set is W[1]-hard, resolving an open problem of Dom et al. [IWPEC, 2008 ]. This is the first bidimensional problem to be shown W[1]-hard. Thus Planar Capacitated Dominating Set can become a useful starting point for reductions showing parameterized intractablility of planar graph problems.

Liar's domination

Abstract: Assume that each vertex of a graph G is the possible location for an “intruder” such as a thief, or a saboteur, a fire in a facility or some possible processor fault in a computer network. A device at a vertex v is assumed to be able to detect the intruder at any vertex in its closed neighborhood N[v] and to identify at which vertex in N[v] the intruder is located. One must then have a dominating set S ⊆ V(G), a set with ∪v∈SN[v] = V(G), to be able to identify any intruder's location. If any one device can fail to detect the intruder, then one needs a double-dominating set. This article introduces the study of liar's dominating sets, sets that can identify an intruder's location even when any one device in the neighborhood of the intruder vertex can lie, that is, any one device in the neighborhood of the intruder vertex can misidentify any vertex in its closed neighborhood as the intruder location. Liar's dominating sets lie between double-dominating sets and triple-dominating sets because every triple-dominating set is a liar's dominating set, and every liar's dominating set must double dominate. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

Reoptimization of Weighted Graph and Covering Problems

Abstract: Given an instance of an optimization problem and a good solution of that instance, the reoptimization is a concept of analyzing how does the solution change when the instance is locally modified. We investigate reoptimization of the following problems: Maximum Weighted Independent Set, Maximum Weighted Clique, Minimum Weighted Dominating Set, Minimum Weighted Set Cover and Minimum Weighted Vertex Cover. The local modifications we consider are addition or removal of a constant number of edges to the graph, or elements to the covering sets in case of Set Cover problem. We present the following results: 1 We provide a PTAS for reoptimization of the unweighted versions of the aforementioned problems when the input solution is optimal. 1 We provide two general techniques for analyzing approximation ratio of the weighted reoptimization problems. 1 We apply our techniques to reoptimization of the considered optimization problems and obtain tight approximation ratios in all the cases.

Hypergraph domination and strong independence

Abstract: We solve several conjectures and open problems from a recent paper by Acharya (2). Some of our results are relatives of the Nordhaus-Gaddum the- orem, concerning the sum of domination parameters in hypergraphs and their complements. (A dominating set in H is a vertex set DX such that, for every vertex x 2 X\D there exists an edge E 2 E with x 2 E and E\D 6 ;.) As an example, it is shown that the tight bound (H)+(H) � n+2 holds in hypergraphs H = (X,E) of order n � 6, where H is defined as H = (X,E) with E = {X \ E | E 2 E}, and is the minimum total cardinality of two disjoint dominating sets. We also present some simple constructions of balanced hypergraphs, disproving conjectures of the aforementioned paper concerning strongly independent sets. (Hypergraph H is balanced if every odd cycle in H has an edge containing three vertices of the cycle; and a set SX is strongly independent if |S \ E| � 1 for all E 2 E.)