Showing papers on "Dominating set published in 2019"

A Dual Attack Detection Technique to Identify Black and Gray Hole Attacks Using an Intrusion Detection System and a Connected Dominating Set in MANETs

Abstract: A mobile ad-hoc network (MANET) is a temporary network of wireless mobile nodes. In a MANET, it is assumed that all of the nodes cooperate with each other to transfer data packets in a multi-hop fashion. However, some malicious nodes don’t cooperate with other nodes and disturb the network through false routing information. In this paper, we propose a prominent technique, called dual attack detection for black and gray hole attacks (DDBG), for MANETs. The proposed DDBG technique selects the intrusion detection system (IDS) node using the connected dominating set (CDS) technique with two additional features; the energy and its nonexistence in the blacklist are also checked before putting the nodes into the IDS set. The CDS is an effective, distinguished, and localized approach for detecting nearly-connected dominating sets of nodes in a small range in mobile ad hoc networks. The selected IDS nodes broadcast a kind of status packet within a size of the dominating set for retrieving the complete behavioral information from their nodes. Later, IDS nodes use our DDBG technique to analyze the collected behavioral information to detect the malicious nodes and add them to the blacklist if the behavior of the node is suspicious. Our experimental results show that the quality of the service parameters of the proposed technique outperforms the existing routing schemes.

Domination When the Stars Are Out

Abstract: We algorithmize the structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is unlikely to be fixed-parameter tractable on the slightly larger class of graphs that exclude K1,4 as an induced subgraph (K1,4-free graphs). We show that our algorithmization can also be used to show that the related Connected Dominating Set problem is fixed-parameter tractable on claw-free graphs. To complement that result, we show that Connected Dominating Set is unlikely to have a polynomial kernel on claw-free graphs and is unlikely to be fixed-parameter tractable on K1,4-free graphs. Combined, our results provide a dichotomy for Dominating Set and Connected Dominating Set on K1,e-free graphs and show that the problem is fixed-parameter tractable if and only if e ≤ 3.

Deterministic Distributed Dominating Set Approximation in the CONGEST Model

Abstract: We develop deterministic approximation algorithms for the minimum dominating set problem in the CONGEST model with an almost optimal approximation guarantee. For e 1/ poly log Δ we obtain two algorithms with approximation factor (1 + e)(1 + l n (Δ + 1)) and with runtimes 2O(√ log n log log n) and O(Δ poly log Δ + poly log Δ log* n), respectively. Further we show how dominating set approximations can be deterministically transformed into a connected dominating set in the CONGEST model while only increasing the approximation guarantee by a constant factor. This results in a deterministic O(log Δ)-approximation algorithm for the minimum connected dominating set with time complexity 2O(√ log n log log n).

On the Parameterized Complexity of Approximating Dominating Set

Abstract: We study the parameterized complexity of approximating the k-Dominating Set (DomSet) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F(k) ⋅ k whenever the graph G has a dominating set of size k. When such an algorithm runs in time T(k) ⋅ poly (n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for k-DomSet. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the “most infamous” open problems in parameterized complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1] ≠ FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T, F and every constant e > 0:• Assuming W[1] ≠ FPT, there is no F(k)-FPT-approximation algorithm for k-DomSet.• Assuming the Exponential Time Hypothesis (ETH), there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ no(k) time.• Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ nk − e time.• Assuming the k-SUM Hypothesis, for every integer k ≥ 3, there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ n⌈ k/2 ⌉ − e time.Previously, only constant ratio FPT-approximation algorithms were ruled out under sf W[1] ≠ FPT and (log1/4 m this allows us to easily apply known techniques to solve them.

More Applications of the d-Neighbor Equivalence: Connectivity and Acyclicity Constraints

Abstract: In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain 2^O(k)* n^O(1), 2^O(k log(k))* n^O(1), 2^O(k^2) * n^O(1) and n^O(k) time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width. For this latter problem, we obtain n^O(k), n^O(k) and n^(2^O(k)) time algorithm parameterized by clique-width, Q-rank-width and rank-width.

Restricted power domination and zero forcing problems

Abstract: Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X. We also study the related problem of finding the smallest zero forcing set that contains a given set of vertices X. The sizes of such sets in a graph G are respectively called the restricted power domination number and restricted zero forcing number of G subject to X. We derive several tight bounds on the restricted power domination and zero forcing numbers of graphs, and relate them to other graph parameters. We also present exact and algorithmic results for computing the restricted power domination number, including integer programs for general graphs and a linear time algorithm for graphs with bounded treewidth. We also use restricted power domination to obtain a parallel algorithm for finding minimum power dominating sets in trees.

Distributed Dominating Set Approximations beyond Planar Graphs

Abstract: The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well known that minimum dominating sets cannot be well approximated locally on general graphs, in recent years there has been much progress on computing good local approximations on sparse graphs and in particular on planar graphs. In this article, we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present(1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and(2) a local O(logan)-time (1+e)-approximation scheme for any e > 0 on graphs of bounded genus.Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. [21]. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments but on combinatorial density arguments only.

How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphs

Abstract: In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarský et al. (J Comput Syst Sci 84:219–242, 2017) proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c, where $$c \ge 1$$ is a fixed integer. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that Vertex Cover admits a polynomial kernel on general graphs for any integer $$c \ge 1$$ , and that Dominating Set does not for any integer $$c \ge 2$$ even on degenerate graphs, unless $$\text {NP} \subseteq \text {coNP}/\text {poly}$$ . For the positive result, we build on the techniques of Jansen and Bodlaender (Proceedings of the 28th symposium on theoretical aspects of computer science (STACS), volume 9 of LIPIcs, pp 177–188, 2011), and for the negative result we use a polynomial parameter transformation for $$c\ge 3$$ and an or-cross-composition for $$c = 2$$ . As existing results imply that Dominating Set admits a polynomial kernel on degenerate graphs for $$c = 1$$ , our result provides a dichotomy about the existence of polynomial kernels for Dominating Set on degenerate graphs with this parameter.

On distance ‐dominating and ‐independent sets in sparse graphs

Abstract: Dvorak (2013) gave a bound on the minimum size of a distance r dominating set in the terms of the maximum size of a distance 2r independent set and generalized coloring numbers, thus obtaining a constant factor approximation algorithm for the parameters in any class of graphs with bounded expansion. We improve and clarify this dependence using an LP-based argument inspired by the work of Bansal and Umboh (2017).

Dominating Sets Construction in RF-based Battery-free Sensor Networks with Full Coverage Guarantee

Abstract: A new network architecture, named as RF-based battery-free sensor network, was proposed in recent years to overcome the lifetime limitation of traditional wireless sensor networks. In an RF-based battery-free sensor network, the battery-free nodes equip no battery and can be recharged by RF-signals. The Dominating Set (DS) is a key method to maintain the coverage of traditional WSNs, and it can be also adopted in the RF-based battery-free sensor networks. However, considering the specific features of RF-based battery-free sensor networks, the DS construction is totally different from that in traditional WSNs. Thus, the problem of constructing DS in a battery-free sensor network is deeply investigated in this article. The NP-Hardness of such problem is proved. Four approximation algorithms are proposed to deal with the snapshot and continuous DS construction requirements, respectively. The approximation ratios of these four algorithms have been analyzed, and the theoretical results show that all these four algorithms are effective. Furthermore, the electromagnetic interference problem in the RF-based battery-free sensor network is considered and defined. An approximated algorithm is proposed to solve such problem. Finally, extensive simulations are carried out. The experimental results verify that the proposed algorithms have high performance in terms of accuracy and efficiency.

The Constant Inapproximability of the Parameterized Dominating Set Problem

Abstract: A set $D$ of vertices of a graph $G$ is a dominating set if every vertex of $G$ is contained in $D$ or adjacent to some vertex of $D$. The number of vertices in a smallest dominating set of $G$ is ...

Adversarially Robust Submodular Maximization under Knapsack Constraints

Abstract: We propose the first adversarially robust algorithm for monotone submodular maximization under single and multiple knapsack constraints with scalable implementations in distributed and streaming settings. For a single knapsack constraint, our algorithm outputs a robust summary of almost optimal (up to polylogarithmic factors) size, from which a constant-factor approximation to the optimal solution can be constructed. For multiple knapsack constraints, our approximation is within a constant-factor of the best known non-robust solution. We evaluate the performance of our algorithms by comparison to natural robustifications of existing non-robust algorithms under two objectives: 1) dominating set for large social network graphs from Facebook and Twitter collected by the Stanford Network Analysis Project (SNAP), 2) movie recommendations on a dataset from MovieLens. Experimental results show that our algorithms give the best objective for a majority of the inputs and show strong performance even compared to offline algorithms that are given the set of removals in advance.

Distributed Dominating Set and Connected Dominating Set Construction Under the Dynamic SINR Model

Abstract: This paper investigates distributed Dominating Set (DS) and Connected Dominating Set (CDS) construction in dynamic wireless networks under the SINR interference model. Specifically, we present a new model for dynamic networks that admits both churns (due to node arrivals/departures) and node mobility. Under this dynamic model, we propose efficient algorithms to construct a DS and a CDS with constant approximation ratios w.r.t. the corresponding minimum ones in O(log n) time with a high probability guarantee. To the best of our knowledge, these algorithms are the first known ones for DS and CDS construction in dynamic networks assuming the SINR interference model. We believe our dynamic network model can greatly facilitate distributed algorithm studies in mobile and dynamic wireless networks.

Semipaired domination in maximal outerplanar graphs

Abstract: A subset S of vertices in a graph G is a dominating set if every vertex in $$V(G) {\setminus } S$$ is adjacent to a vertex in S. If the graph G has no isolated vertex, then a semipaired dominating set of G is a dominating set of G with the additional property that the set S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number $$\gamma _{\mathrm{pr2}}(G)$$ is the minimum cardinality of a semipaired dominating set of G. Let G be a maximal outerplanar graph of order n with $$n_2$$ vertices of degree 2. We show that if $$n \ge 5$$ , then $$\gamma _{\mathrm{pr2}}(G) \le \frac{2}{5}n$$ . Further, we show that if $$n \ge 3$$ , then $$\gamma _{\mathrm{pr2}}(G) \le \frac{1}{3}(n+n_2)$$ . Both bounds are shown to be tight.

Vertex-edge domination in graphs

Abstract: We establish that for any connected graph G of order $$n \ge 6$$ , a minimum vertex-edge dominating set of G has at most n/3 vertices, thus affirmatively answering the open question posed by Boutrig et al. (Aequ Math 90(2):355–366, 2016).

Deterministic Distributed Dominating Set Approximation in the CONGEST Model

Abstract: We develop deterministic approximation algorithms for the minimum dominating set problem in the CONGEST model with an almost optimal approximation guarantee. For $\epsilon>1/{\text{{poly}}}\log \Delta$ we obtain two algorithms with approximation factor $(1+\epsilon)(1+\ln (\Delta+1))$ and with runtimes $2^{O(\sqrt{\log n \log\log n})}$ and $O(\Delta\cdot\text{poly}\log \Delta +\text{poly}\log \Delta \log^{*} n)$, respectively. Further we show how dominating set approximations can be deterministically transformed into a connected dominating set in the \CONGEST model while only increasing the approximation guarantee by a constant factor. This results in a deterministic $O(\log \Delta)$-approximation algorithm for the minimum connected dominating set with time complexity $2^{O(\sqrt{\log n \log\log n})}$.

Graph Theory Approach to the Vulnerability of Transportation Networks

Abstract: Nowadays, transport is the basis for the functioning of national, continental, and global economies. Thus, many governments recognize it as a critical element in ensuring the daily existence of societies in their countries. Those responsible for the proper operation of the transport sector must have the right tools to model, analyze, and optimize its elements. One of the most critical problems is the need to prevent bottlenecks in transport networks. Thus, the main aim of the article was to define the parameters characterizing the transportation network vulnerability and select algorithms to support their search. The parameters proposed are based on characteristics related to domination in graph theory. The domination, edge-domination concepts, and related topics, such as bondage-connected and weighted bondage-connected numbers, were applied as the tools for searching and identifying the bottlenecks in transportation networks. Furthermore, the algorithms for finding the minimal dominating set and minimal (maximal) weighted dominating sets are proposed. This way, the exemplary academic transportation network was analyzed in two cases: stationary and dynamic. Some conclusions are presented. The main one is the fact that the methods given in this article are universal and applicable to both small and large-scale networks. Moreover, the approach can support the dynamic analysis of bottlenecks in transport networks.

Hop Dominating Sets in Graphs Under Binary Operations

Abstract: Let G be a (simple) connected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a hop dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2. The minimum cardinality of a hop dominating set of G, denoted by γh(G), is called the hop domination number of G. In this paper we revisit the concept of hop domination, relate it with other domination concepts, and investigate it in graphs resulting from some binary operations.

Energy-Efficient Sleep Scheduling in WBANs: From the Perspective of Minimum Dominating Set

Abstract: Wireless body area networks (WBANs) that offer various medical applications have received considerable attention in recent years. Due to limited energy of sensors, duty-cycling technique is employed to prolong the network lifetime. However, it results in long delivery delay and suffers from reliability issues. In this paper, we introduce an efficient and reliable sleep scheduling scheme from the perspective of constructing ${m}$ -fold dominating set (DS), where ${m}$ is the number of links from a node outside DS to those in DS. The key idea is to activate partial nodes at each frame to form a DS which can guarantee the network reliability such that the other nodes can fall asleep to save energy. Technically, we formulate the sleep scheduling in a WBAN as a problem of constructing minimum weighted ${m}$ -fold DS, which is proven NP-hard. We first design an ${H}$ ( ${m}\,\,\boldsymbol {+}\,\,\boldsymbol {\delta }$ )-approximation algorithm, namely global approximation algorithm, by globally picking the optimal node based on a polymatroid function, where ${H}(\boldsymbol \cdot)$ is the Harmonic number and $\boldsymbol \delta $ is the maximum node degree. Then, we propose a simplified $1+\ln (m\delta)$ -approximation algorithm, referred to as local approximation algorithm, to reduce computational complexity and execution rounds. We further conduct extensive simulations to confirm the superiority of our proposed algorithms.