scispace - formally typeset
Search or ask a question

Showing papers on "Dominating set published in 2016"


Journal ArticleDOI
TL;DR: The first polylogarithmic lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching is given.
Abstract: The question of what can be computed, and how efficiently, is at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a distributed fashion. More precisely, if nodes of a network must base their decision on information in their local neighborhood only, how well can they compute or approximate a global (optimization) problem? In this paper we give the first polylogarithmic lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching. In addition, we present a new distributed algorithm for solving general covering and packing linear programs. For some problems this algorithm is tight with the lower bounds, whereas for others it is a distributed approximation scheme. Together, our lower and upper bounds establish the local computability and approximability of a large class of problems, characterizing how much local information is required to solve these tasks.

177 citations


Journal ArticleDOI
TL;DR: The field of exact exponential time algorithms for non-deterministic polynomial-time hard problems has thrived since the mid-2000s as discussed by the authors, and exhaustive search remains asymptotically the fastest known algorithm for some basic problems.
Abstract: The field of exact exponential time algorithms for non-deterministic polynomial-time hard problems has thrived since the mid-2000s. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set, and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2n), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2n), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every e

106 citations


Journal ArticleDOI
TL;DR: It is proved that if $G$ is an $n$-vertex isolate-free graph with $\ell$ vertices of degree 1, then $\gamma_g(G) \le 3n/5 + \left \lceil \ell/2 \right \rceil + 1$; in the course of establishing this result, a question of Bresar e...
Abstract: In the domination game on a graph $G$, the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices dominated. This process eventually produces a dominating set of $G$; Dominator aims to minimize the size of this set, while Staller aims to maximize it. The size of the dominating set produced under optimal play is the game domination number of $G$, denoted by $\gamma_g (G)$. In this paper, we prove that $\gamma_g(G) \le 2n/3$ for every $n$-vertex isolate-free graph $G$. When $G$ has minimum degree at least $2$, we prove the stronger bound $\gamma_g(G) \le 3n/5$; this resolves a special case of a conjecture due to Kinnersley, West, and Zamani [SIAM J. Discrete Math., 27 (2013), pp. 2090--2107]. Finally, we prove that if $G$ is an $n$-vertex isolate-free graph with $\ell$ vertices of degree 1, then $\gamma_g(G) \le 3n/5 + \left \lceil \ell/2 \right \rceil + 1$; in the course of establishing this result, we answer a question of Bresar e...

57 citations


Proceedings ArticleDOI
01 Jan 2016
TL;DR: Fomin et al. as mentioned in this paper showed that for every positive integer r and for every graph class G of bounded expansion, the r-dominating set problem admits a linear kernel on graphs from G.
Abstract: We prove that for every positive integer r and for every graph class G of bounded expansion, the r-DOMINATING SET problem admits a linear kernel on graphs from G. Moreover, in the more general case when G is only assumed to be nowhere dense, we give an almost linear kernel on G for the classic DOMINATING SET problem, i.e., for the case r=1. These results generalize a line of previous research on finding linear kernels for DOMINATING SET and r-DOMINATING SET (Alber et al., JACM 2004, Bodlaender et al., FOCS 2009, Fomin et al., SODA 2010, Fomin et al., SODA 2012, Fomin et al., STACS 2013). However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches. We complement our findings by showing that for the closely related CONNECTED DOMINATING SET problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on H-topological-minor-free graphs (Fomin et al., STACS 2013). Also, we prove that for any somewhere dense class G, there is some r for which r-DOMINATING SET is W[2]-hard on G. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of r-DOMINATING SET on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.

44 citations


Journal ArticleDOI
TL;DR: It is proved that the problem remains PSPACE-complete even for planar graphs, bounded bandwidth graphs, split graphs, and bipartite graphs and the problem can be solved in linear time for cographs, forests, and interval graphs.

43 citations


Journal ArticleDOI
TL;DR: A number of models for mobile guards on the vertices of a graph when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover are described.
Abstract: Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.

42 citations


Journal ArticleDOI
TL;DR: This paper designs an effective hybrid memetic algorithm (HMA) for the minimum weight-dominating set problem, which contains a greedy randomized adaptive construction procedure, a tabu local search procedure, an crossover operator, a population-updating method, and a path-relinking procedure.
Abstract: The minimum weight-dominating set (MWDS) problem is NP-hard and has a lot of applications in the real world. Several metaheuristic methods have been developed for solving the problem effectively, but suffering from high CPU time on large-scale instances. In this paper, we design an effective hybrid memetic algorithm (HMA) for the MWDS problem. First, the MWDS problem is formulated as a constrained 0–1 programming problem and is converted to an equivalent unconstrained 0–1 problem using an adaptive penalty function. Then, we develop a memetic algorithm for the resulting problem, which contains a greedy randomized adaptive construction procedure, a tabu local search procedure, a crossover operator, a population-updating method, and a path-relinking procedure. These strategies make a good tradeoff between intensification and diversification. A number of experiments were carried out on three types of instances from the literature. Compared with existing algorithms, HMA is able to find high-quality solutions in much less CPU time. Specifically, HMA is at least six times faster than existing algorithms on the tested instances. With increasing instance size, the CPU time required by HMA increases much more slowly than required by existing algorithms.

42 citations



Journal ArticleDOI
TL;DR: It is shown that all minimal dominating sets of a chordal bipartite graph can be generated in incremental polynomial, hence output polynometric, time, and this result contributes to the known cases having an affirmative reply to this important question.

36 citations


Journal ArticleDOI
TL;DR: A generalized power domination problem, which limits the number of propagation iterations to a given positive integer; that is, the second rule is applied synchronously with a bounded time constraint, and a linear time algorithm that uses a labeling approach is proposed.
Abstract: Based on the power observation rules, the problem of monitoring a power utility network can be transformed into the graph-theoretic power domination problem, which is an extension of the well-known domination problem. A set $$S$$S is a power dominating set (PDS) of a graph $$G=(V,E)$$G=(V,E) if every vertex $$v$$v in $$V$$V can be observed under the following two observation rules: (1) $$v$$v is dominated by $$S$$S, i.e., $$v \in S$$v?S or $$v$$v has a neighbor in $$S$$S; and (2) one of $$v$$v's neighbors, say $$u$$u, and all of $$u$$u's neighbors, except $$v$$v, can be observed. The power domination problem involves finding a PDS with the minimum cardinality in a graph. Similar to message passing protocols, a PDS can be considered as a dominating set with propagation that applies the second rule iteratively. This study investigates a generalized power domination problem, which limits the number of propagation iterations to a given positive integer; that is, the second rule is applied synchronously with a bounded time constraint. To solve the problem in block graphs, we propose a linear time algorithm that uses a labeling approach. In addition, based on the concept of time constraints, we provide the first nontrivial lower bound for the power domination problem.

32 citations


Journal ArticleDOI
TL;DR: By proving that an optimal solution has a specific decomposition, it is proved that the approximation ratio is$$\alpha +2(1+\ln \alpha )$$α+2 (1+lnα), where $$\alpha $$α is the approximation ratios for the minimum $$(1,m)$$(2,m)-CDS problem.
Abstract: To save energy and alleviate interference in a wireless sensor network, connected dominating set (CDS) has been proposed as the virtual backbone. Since nodes may fail due to accidental damage or energy depletion, it is desirable to construct a fault tolerant CDS, which can be modeled as a $$k$$k-connected $$m$$m-fold dominating set $$((k,m)$$((k,m)-CDS for short): a subset of nodes $$C\subseteq V(G)$$C⊆V(G) is a $$(k,m)$$(k,m)-CDS of $$G$$G if every node in $$V(G)\setminus C$$V(G)\C is adjacent with at least $$m$$m nodes in $$C$$C and the subgraph of $$G$$G induced by $$C$$C is $$k$$k-connected.In this paper, we present an approximation algorithm for the minimum $$(2,m)$$(2,m)-CDS problem with $$m\ge 2$$m?2. Based on a $$(1,m)$$(1,m)-CDS, the algorithm greedily merges blocks until the connectivity is raised to two. The most difficult problem in the analysis is that the potential function used in the greedy algorithm is not submodular. By proving that an optimal solution has a specific decomposition, we managed to prove that the approximation ratio is $$\alpha +2(1+\ln \alpha )$$?+2(1+ln?), where $$\alpha $$? is the approximation ratio for the minimum $$(1,m)$$(1,m)-CDS problem. This improves on previous approximation ratios for the minimum $$(2,m)$$(2,m)-CDS problem, both in general graphs and in unit disk graphs.

Journal ArticleDOI
15 May 2016
TL;DR: This paper proposes a new degree-based greedy approximation algorithm named as Connected Pseudo Dominating Set Using 2 Hop Information (CPDS2HI), which reduces the CDS size as much as possible and is the most time efficient and size-optimal CDS construction algorithm.
Abstract: In a wireless network, messages need to be sent on in an optimized way to preserve the energy of the network. A minimum connected dominating set (MCDS) offers an optimized way of sending messages. However, MCDS construction is a NP-Hard problem. In this paper, we propose a new degree-based greedy approximation algorithm named as Connected Pseudo Dominating Set Using 2 Hop Information (CPDS2HI), which reduces the CDS size as much as possible. Our method first constructs the CDS and then reduces its size further by excluding some of the CDS nodes cleverly without any loss in coverage or connectivity. The simulation results show that our method outperforms existing CDS construction algorithms in terms of both the CDS size and construction cost. CPDS2HI retains the current best performance ratio of ( 4.8 + ln 5 ) | o p t | + 1.2 , |opt| being the size of an optimal CDS of the network, and has the best time complexity of O(D), where D is the network diameter. To the best of our knowledge this is the most time efficient and size-optimal CDS construction algorithm. It has a linear message complexity of O(nΔ), where n is the network size and Δ is the maximum degree of all the nodes.

Proceedings ArticleDOI
10 Apr 2016
TL;DR: This is the first performance-guaranteed algorithm for the minimum (3, m)-CDS problem in a general wireless network, and improves previous performance ratio in a homogeneous wireless sensor network by a large amount.
Abstract: Using a connected dominating set (CDS) to serve as a virtual backbone of a wireless sensor network is an effective way to save energy and alleviate broadcasting storm. Since nodes may fail due to accidental damage or energy depletion, it is desirable to construct a fault tolerant CDS, which can be modeled as a k-connected m-fold dominating set ((k, m)-CDS for short). A subset of nodes C ⊆ V(G) is a (k, m)-CDS of G if every node in V(G)\C is adjacent with at least m nodes in C and the subgraph of G induced by C is k-connected. In this paper, we present an approximation algorithm for the minimum (3, m)-CDS problem with m > 3, which has size at most γ times that of an optimal solution, where γ = α + 8 + 21n(2α — 6) for α > 4 and γ = 3α + 2 In 2 for α < 4, and α is the approximation ratio for the minimum (2, m)-CDS problem. This is the first performance-guaranteed algorithm for the minimum (3, m)-CDS problem in a general wireless network, and improves previous performance ratio in a homogeneous wireless sensor network by a large amount.

Proceedings ArticleDOI
25 Jul 2016
TL;DR: A deterministic and local constant factor approximation for minimum dominating sets on bounded genus graphs, a large family of sparse graphs using a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al.
Abstract: The Minimum Dominating Set (MDS) problem is not only one of the most fundamental problems in distributed computing, it is also one of the most challenging ones. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, several breakthroughs have been made on computing local approximations on sparse graphs.This paper presents a deterministic and local constant factor approximation for minimum dominating sets on bounded genus graphs, a large family of sparse graphs. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments. We believe that our techniques can be useful for the study of local problems on sparse graphs beyond the scope of this paper.

Journal ArticleDOI
TL;DR: It is proved that γ ( G ) ?

Journal ArticleDOI
TL;DR: An efficient randomized iterated greedy approach for the minimum weight dominating set problem, where—given a vertex-weighted graph—the goal is to identify a subset of the graphs’ vertices with minimum total weight such that each vertex of the graph is either in the subset or has a neighbor in the subsets.

Journal ArticleDOI
TL;DR: It is proved that by adopting a greedy strategy, Dominator can complete the total domination game played in a graph with minimum degree at least 2 in at most 3 n / 4 moves.

Journal ArticleDOI
TL;DR: This paper studies a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G), and the total domination number: γt(G).
Abstract: Abstract Let G be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G), and the total domination number, γt(G). A set S of vertices in a graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, γt2(G), is the minimum cardinality of a semitotal dominating set of G. We observe that γ(G) ≤ γt2(G) ≤ γt(G). We characterize the set of vertices that are contained in all, or in no minimum semitotal dominating set of a tree.

Journal ArticleDOI
TL;DR: This paper considers two covering location problems on a network where the demand is distributed along the edges, the classical maximal covering location problem and the obnoxious version where the coverage should be minimized subject to some distance constraints between the facilities.

Proceedings ArticleDOI
01 Oct 2016
TL;DR: In this article, it was shown that there is no FPT algorithm that can approximate the dominating set problem with any constant ratio, unless FPT = W[1] and the hardness reduction was built on the second author's recent W[ 1]-hardness proof of the biclique problem.
Abstract: We prove that there is no fpt-algorithm that can approximate the dominating set problem with any constant ratio, unless FPT = W[1]. Our hardness reduction is built on the second author's recent W[1]-hardness proof of the biclique problem [25]. This yields, among other things, a proof without the PCP machinery that the classical dominating set problem has no polynomial time constant approximation under the exponential time hypothesis.

Journal ArticleDOI
TL;DR: This paper presents a linear-time algorithm for k-power domination in block graphs, a common generalization of domination and power domination, and presents a solution to the power system monitoring problem.
Abstract: The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating set in graphs. The concept of $$k$$k-power domination, first introduced by Chang et al. (Discret Appl Math 160:1691---1698, 2012), is a common generalization of domination and power domination. In this paper, we present a linear-time algorithm for $$k$$k-power domination in block graphs.

Proceedings ArticleDOI
01 Aug 2016
TL;DR: This paper designs the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks), and settles a conjecture of Har-Peled in the affirmative.
Abstract: In this paper, we design the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks) We show that the local search algorithm that yields PTASs when the regions are disks [Aschner/Katz/Morgenstern/Yuditsky, WALCOM 2013; Gibson/Pirwani, 2005; Mustafa/Raman/Ray, 2015] can be extended to work for non-piercing regions While such an extension is intuitive and natural, attempts to settle this question have failed even for pseudodisks The techniques used for analysis when the regions are disks rely heavily on the underlying geometry, and do not extend to topologically defined settings such as pseudodisks In order to prove our results, we introduce novel techniques that we believe will find applications in other problems We then consider the Capacitated Region Packing problem Here, the input consists of a set of points with capacities, and a set of regions The objective is to pick a maximum cardinality subset of regions so that no point is covered by more regions than its capacity We show that this problem admits a PTAS when the regions are k-admissible regions (pseudodisks are 2-admissible), and the capacities are bounded Our result settles a conjecture of Har-Peled (see Conclusion of [Har-Peled, SoCG 2014]) in the affirmative The conjecture was for a weaker version of the problem, namely when the regions are pseudodisks, the capacities are uniform, and the point set consists of all points in the plane Finally, we consider the Capacitated Point Packing problem In this setting, the regions have capacities, and our objective is to find a maximum cardinality subset of points such that no region has more points than its capacity We show that this problem admits a PTAS when the capacity is unity, extending one of the results of Ene et al [Ene/Har-Peled/Raichel, SoCG 2012]

Journal ArticleDOI
TL;DR: This work completely determine the complexity of the dominating set problem for classes defined by forbidding a five-vertex path and any set of fragments with at most five vertices and proves polynomial-time solvability of the problem for some two classes of a similar type.
Abstract: We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the complexity of the problem even for hereditary classes with small forbidden structures. We completely determine the complexity of the problem for classes defined by forbidding a five-vertex path and any set of fragments with at most five vertices. Additionally, we also prove polynomial-time solvability of the problem for some two classes of a similar type. The notion of a boundary class is a helpful tool for analyzing the computational complexity of graph problems in the family of hereditary classes. Three boundary classes were known for the dominating set problem prior to this paper. We present a new boundary class for it.

Journal ArticleDOI
TL;DR: The enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality is studied and enumeration algorithms as well as lower and upper bounds for the maximum number of minimalconnected dominating sets in such graphs are established.

Journal ArticleDOI
TL;DR: It is proved that if all roots of D ( G , x ) are real, then ?

Journal ArticleDOI
TL;DR: Simulation results indicated that the wireless sensor network embedded with Hopfield neural network as a static optimizer performed competitively with other local or distributed algorithms for the weakly connected dominating set problem to establish its feasibility.
Abstract: This paper proposes embedding an artificial neural network into a wireless sensor network in fully parallel and distributed computation mode. The goal is to equip the wireless sensor network with computational intelligence and adaptation capability for enhanced autonomous operation. The applicability and utility of the proposed concept is demonstrated through a case study whereby a Hopfield neural network configured as a static optimizer for the weakly-connected dominating set problem is embedded into a wireless sensor network to enable it to adapt its network infrastructure to potential changes on-the-fly and following deployment in the field. Minimum weakly-connected dominating set defined for the graph model of the wireless sensor network topology is employed to represent the network infrastructure and can be recomputed each time the sensor network topology changes. A simulation study using the TOSSIM emulator for TinyOS-Mica sensor network platform was performed for mote counts of up to 1000. Time complexity, message complexity and solution quality measures were assessed and evaluated for the case study. Simulation results indicated that the wireless sensor network embedded with Hopfield neural network as a static optimizer performed competitively with other local or distributed algorithms for the weakly connected dominating set problem to establish its feasibility.

Journal ArticleDOI
TL;DR: In this paper, the authors studied finite simple graphs with dominating induced matchings, i.e. an induced matching which forms a maximal matching, and showed that the induced matching number of such graphs is at most the minimum matching number.
Abstract: The regularity of the edge ideal of a finite simple graph G is at least the induced matching number of G and is at most the minimum matching number of G. If G possesses a dominating induced matching, i.e. an induced matching which forms a maximal matching, then the induced matching number of G is equal to the minimum matching number of G. In the present paper, from viewpoints of both combinatorics and commutative algebra, finite simple graphs with dominating induced matchings will be mainly studied.

Journal ArticleDOI
TL;DR: In this study, the problem of building cluster-based topologies for Wireless Sensor Networks with several sinks is considered and several optimization criteria are proposed to implicitly or explicitly balance the topology.

Journal ArticleDOI
TL;DR: The task of complete complexity dichotomy is to clearly distinguish between easy and hard cases of a given problem on a family of subproblems for some optimization problems restricted to certain classes of graphs closed under deletion of vertices.
Abstract: The task of complete complexity dichotomy is to clearly distinguish between easy and hard cases of a given problem on a family of subproblems. We consider this task for some optimization problems restricted to certain classes of graphs closed under deletion of vertices. A concept in the solution process is based on revealing the so-called “critical” graph classes, which play an important role in the complexity analysis for the family. Recent progress in studying such classes is presented in the article.

Journal ArticleDOI
TL;DR: In this paper, an improved algorithm for the maximum independent set problem in an n -vertex graph with degree bounded by 5 is presented, improving the previous running time bound of 1.189 5 n n O ( 1 ) .