Showing papers on "Dominating set published in 2017"
14 Aug 2017
TL;DR: In particular, Chen et al. as mentioned in this paper showed that there is no O(opt)-FPT-approximation algorithm for Clique and no f-opt-FPT algorithm for DomSet, for any function f (e.g., this holds even if f is an exponential or the Ackermann function).
Abstract: We consider questions that arise from the intersection between theareas of approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx, 2008; Fellow et al., 2012; Downey & Fellow 2013]) are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting opt be the optimum and N be the size of the input, is there an algorithm that runs int(opt) poly(N) time and outputs a solution of size f(opt), forany functions t and f that are independent of N (for Clique, we want f(opt)=Ω(1))? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no o(opt)-FPT-approximation algorithm for Clique and no f(opt)-FPT-approximation algorithm for DomSet, for any function f (e.g., this holds even if f is an exponential or the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur, 2016, Manurangsi & Raghavendra 2016], which states that no 2^{o(n)}-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 - c)-satisfiable for some constant c ≈ 0.Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs. Previously only exact versions of these problems were known to be W[1]-hard [Lin, 2015; Khot & Raman, 2000; Moser & Sikdar, 2009]. Additionally, we rule out k^{o(1)}-FPT-approximation algorithm for Densest k-Subgraph although this ratio does not yet match the trivial O(k)-approximation algorithm.To the best of our knowledge, prior results only rule out constantfactor approximation for Clique [Hajiaghayi et al., 2013; KK13, Bonnet et al., 2015] and log^{1/4+c}(opt) approximation for DomSet for any constant c ≈ 0 [Chen & Lin, 2016]. Our result on Clique significantly improves on [Hajiaghayi et al., 2013; Bonnet et al., 2015]. However, our result on DomSet is incomparable to [Chen & Lin, 2016] since their results hold under ETH while our results hold under Gap-ETH, which is a stronger assumption.
61 citations
14 Nov 2017
TL;DR: In this paper, the authors investigated the family of intersection graphs of low density objects in low dimensional Euclidean space, which includes planar graphs and has some interesting properties, and in particular, it is a subset of the class of graphs that have polynomial expansion.
Abstract: We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. This family of graphs has some interesting properties, and in particular, it is a subset of the family of graphs that have polynomial expansion.
52 citations
TL;DR: A new local search algorithm for the Minimum Weight Dominating Set problem is proposed, which is based on a heuristic called two-level configuration checking (CC2), which is a new variant of a recent powerful configuration checking strategy (CC) for effectively avoiding the recent search paths.
Abstract: The Minimum Weight Dominating Set (MWDS) problem is an important generalization of the Minimum Dominating Set (MDS) problem with extensive applications. This paper proposes a new local search algorithm for the MWDS problem, which is based on two new ideas. The first idea is a heuristic called two-level configuration checking (CC2), which is a new variant of a recent powerful configuration checking strategy (CC) for effectively avoiding the recent search paths. The second idea is a novel scoring function based on the frequency of being uncovered of vertices. Our algorithm is called CC2FS, according to the names of the two ideas. The experimental results show that, CC2FS performs much better than some state-of-the-art algorithms in terms of solution quality on a broad range of MWDS benchmarks.
47 citations
01 Jan 2017
TL;DR: In this paper, it was shown that for every fixed integer r, the parameterized distance dominating set problem admits an almost linear kernel on any nowhere dense graph class, and that the limit of parameterized tractability of DDS on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.
Abstract: We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r,eps)|A|^(1+eps), where r is any positive integer, eps is any positive real, and f is a function that depends only on the class C. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by [Reidl et al., CoRR, 2016]. As an algorithmic application of the above result, we show that for every fixed integer r, the parameterized Distance-r Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by [Drange et al., STACS 2016], and shows that the limit of parameterized tractability of Distance-r Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.
46 citations
TL;DR: This paper studies Upper Domination, i.e., the problem of computing the maximum cardinality of a minimal dominating set in a graph with respect to classical and parameterised complexity as well as approximability.
43 citations
Posted Content•
TL;DR: In this paper, the authors studied the parameterized complexity of approximating the dominating set problem and showed that there is no FPT-approximation algorithm for the problem with complexity at most O(k) for any constant k > 0.
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 $F(k) \cdot k$ whenever the graph $G$ has a dominating set of size $k$. When such an algorithm runs in time $T(k) \cdot poly(n)$ (i.e., FPT-time) for some computable function $T$, it is said to be an $F(k)$-FPT-approximation algorithm for $k$-DomSet. We prove the following for every computable functions $T, F$ and every constant $\varepsilon > 0$:
$\bullet$ Assuming $W[1]
eq FPT$, there is no $F(k)$-FPT-approximation algorithm for $k$-DomSet.
$\bullet$ Assuming the Exponential Time Hypothesis (ETH), there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{o(k)}$ time.
$\bullet$ Assuming the Strong Exponential Time Hypothesis (SETH), for every integer $k \geq 2$, there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{k - \varepsilon}$ time.
$\bullet$ Assuming the $k$-Sum Hypothesis, for every integer $k \geq 3$, there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{\lceil k/2 \rceil - \varepsilon}$ time.
Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well studied problem or a variant of one; this allows us to easily apply known techniques to solve them.
35 citations
TL;DR: In this paper, the authors presented a constant approximation algorithm for the minimum weight (k,m)$ -CDS problem in unit disk graphs under the assumption that $k$ and $m$ are two fixed constants with $m\geq k$.
Abstract: In a wireless sensor network, the virtual backbone plays an important role. Due to accidental damage or energy depletion, it is desirable that the virtual backbone is fault-tolerant. A fault-tolerant virtual backbone can be modeled as a $k$ -connected $m$ -fold dominating set ( $(k,m)$ -CDS for short). In this paper, we present a constant approximation algorithm for the minimum weight $(k,m)$ -CDS problem in unit disk graphs under the assumption that $k$ and $m$ are two fixed constants with $m\geq k$ . Prior to this paper, constant approximation algorithms are known for $k=1$ with weight and $2\leq k\leq 3$ without weight. Our result is the first constant approximation algorithm for the $(k,m)$ -CDS problem with general $k,m$ and with weight. The performance ratio is $(\alpha +5\rho )$ for $k\geq 3$ and $(\alpha +2.5\rho )$ for $k=2$ , where $\alpha $ is the performance ratio for the minimum weight $m$ -fold dominating set problem and $\rho $ is the performance ratio for the subset $k$ -connected subgraph problem (both problems are known to have constant performance ratios).
35 citations
16 Jan 2017
TL;DR: Tools from logic are used from a sub-field of model theory known as stability theory to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness, which implies that for every class C of graphs, the distance-r dominating set problem admits a kernel on C for every value of r.
Abstract: Nowhere dense classes of graphs [21, 22] are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness, a concept originating in finite model theory, has proved to be particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on nowhere dense classes. However, the existing constructions showing the equivalence of nowhere denseness and uniform quasi-wideness imply a non-elementary blow up in the parameter dependence of the fpt-algorithms, making them infeasible in practice. As a first main result of this article, we use tools from logic, in particular from a sub-field of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness. A powerful method in parameterized complexity theory is to compute a problem kernel in a pre-computation step, that is, to reduce the input instance in polynomial time to a sub-instance of size bounded in the parameter only (independently of the input graph size). Our new tools allow us to obtain for every fixed radius r i N a polynomial kernel for the distance-r dominating set problem on nowhere dense classes of graphs. This result is particularly interesting, as it implies that for every class C of graphs that is closed under taking subgraphs, the distance-r dominating set problem admits a kernel on C for every value of r if, and only if, it already admits a polynomial kernel for every value of r (under the standard assumption of parameterized complexity theory that FPT ≠ W[2]).
31 citations
TL;DR: This article considers kernelization for problems on d-degenerate graphs, that is, graphs such that any subgraph contains a vertex of degree at most d, and proves that unless coNP ⊆ NP/poly Dominating Set has no kernels of size O(kd−1)(d−3)−ε) for any ε > 0.
Abstract: Kernelization is a strong and widely applied technique in parameterized complexity. In a nutshell, a kernelization algorithm for a parameterized problem transforms in polynomial time a given instance of the problem into an equivalent instance whose size depends solely on the parameter. Recent years have seen major advances in the study of both upper and lower bound techniques for kernelization, and by now this area has become one of the major research threads in parameterized complexity.In this article, we consider kernelization for problems on d-degenerate graphs, that is, graphs such that any subgraph contains a vertex of degree at most d. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, for example, planar graphs, H-minor free graphs, and H-topological-minor free graphs. We show that for several natural problems on d-degenerate graphs the best-known kernelization upper bounds are essentially tight. In particular, using intricate constructions of weak compositions, we prove that unless coNP ⊆ NP/poly:b Dominating Set has no kernels of size O(k(d−1)(d−3)−e) for any e > 0. The current best upper bound is O(k(d+1)2).b Independent Dominating Set has no kernels of size O(kd−4−e) for any e > 0. The current best upper bound is O(kd+1).b Induced Matching has no kernels of size O(kd−3−e) for any e > 0. The current best upper bound is O(kd).To the best of our knowledge, Dominating Set is the the first problem where a lower bound with superlinear dependence on d (in the exponent) can be proved.In the last section of the article, we also give simple kernels for Connected Vertex Cover and Capacitated Vertex Cover of size O(kd) and O(kd+1), respectively. We show that the latter problem has no kernels of size O(kd−e) unless coNP ⊆ NP/poly by a simple reduction from d-Exact Set Cover (the same lower bound for Connected Vertex Cover on d-degenerate graphs is already known).
28 citations
TL;DR: This paper studies the minimum partial positive influence dominating set (MPPIDS) problem, and presents an approximation algorithm for a more general problem called minimum partial set multicover problem, which has a constant performance ratio.
Abstract: Influence problem is one of the central problems in the study of online social networks, the goal of which is to influence all nodes with the minimum number of seeds. However, in the real world, it might be too expensive to influence all nodes. In many cases, it is satisfactory to influence nodes only up to some percent p. In this paper, we study the minimum partial positive influence dominating set (MPPIDS) problem. In fact, we presented an approximation algorithm for a more general problem called minimum partial set multicover problem. As a consequence, the MPPIDS problem admits an approximation with performance ratio $$\gamma H(\Delta )$$źH(Δ), where $$H(\cdot )$$H(·) is the Harmonic number, $$\gamma =1/(1-(1-p)\eta ),\eta \approx \Delta ^2/\delta $$ź=1/(1-(1-p)ź),źźΔ2/ź, and $$\Delta ,\delta $$Δ,ź are the maximum degree and the minimum degree of the graph, respectively. For power-law graphs, we show that our algorithm has a constant performance ratio.
27 citations
TL;DR: An LP rounding algorithm is given that achieves an approximation factor of 3a and is complemented with a matching (up to constants) hardness of approximation result.
TL;DR: In this paper, the authors study the game of total domination in trees and show that if a tree is a forest with no isolated vertex, then the game total domination number γ tg (G ) ≤ γ ng(G ) of G is the number of vertices chosen when Dominator and Staller start the game and both players play optimally.
01 Aug 2017
TL;DR: A distributed approximation named CS-Cluster is proposed to address the MCDS problem under unit disk graph, which is proved to be NP-complete, and the approximation ratio is a great improvement.
Abstract: Wireless sensor networks (WSNs) have been widely used in a plenty of applications. To achieve higher efficiency for data collection, WSNs are often partitioned into several disjointed clusters, each with a representative cluster head in charge of the data gathering and routing process. Such a partition is balanced and effective, if the distance between each node and its cluster head can be bounded within a constant number of hops, and any two cluster heads are connected. Finding such a cluster partition with minimum number of clusters and connectors between cluster heads is defined as minimum connected $d$ -hop dominating set ( $d$ -MCDS) problem, which is proved to be NP-complete. In this paper, we propose a distributed approximation named CS-Cluster to address the $d$ -MCDS problem under unit disk graph . CS-Cluster constructs a sparser $d$ -hop maximal independent set ( $d$ -MIS), connects the $d$ -MIS, and finally checks and removes redundant nodes. We prove the approximation ratio of CS-Cluster is $(2d+1)\lambda $ , where $\lambda $ is a parameter related with $d$ but is no more than 18.4. Compared with the previous best result $O(d^{2})$ , our approximation ratio is a great improvement. Our evaluation results demonstrate the outstanding performance of our algorithm compared with previous works.
TL;DR: This paper presents a subexponential algorithm for the Maximum (Weighted) Independent Set Problem in P t -free graphs and states that the complexity status for larger t is still unknown.
TL;DR: A greedy randomized adaptive search procedure with path cost heuristic for MIDS, as well as the classical tabu mechanism, which makes better use of the vertex neighborhood information provided by path cost and thus is able to discover better and more solutions and to escape from local optimal solutions when the original GRASP fails to find new improved solutions.
Abstract: The minimum independent dominating set problem (MIDS) is an extension of the classical dominating set problem with wide applications. In this paper, we describe a greedy randomized adaptive search procedure (GRASP) with path cost heuristic for MIDS, as well as the classical tabu mechanism. Our novel GRASP algorithm makes better use of the vertex neighborhood information provided by path cost and thus is able to discover better and more solutions and to escape from local optimal solutions when the original GRASP fails to find new improved solutions. Moreover, to further overcome the serious cycling problem, the tabu mechanism is employed to forbid some just-removed vertices back to the candidate solution. Computational experiments carried out on standard benchmarks, namely DIMACS instances, show that our algorithm consistently outperforms two MIDS solvers as well as the original GRASP.
TL;DR: It is proved that almost all graphs have a hop dominating set of cardinality at most the total domination number if p is constant, and that the decision problems for the 2-step dominating set andHop dominating set problems are NP-complete for planar bipartite graphs and planar chordal graphs.
Abstract: Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G is a 2-step dominating set of G if every vertex is 2-step dominated by some vertex of S. A subset S of vertices of G is a hop dominating set if every vertex outside S is 2-step dominated by some vertex of S. The hop domination number, $$\gamma _{h}(G)$$
, of G is the minimum cardinality of a hop dominating set of G. It is known that for a connected graph G, $$\gamma _{h}(G) = |V(G)|$$
if and only if G is a complete graph. We characterize the connected graphs G for which $$\gamma _{h}(G) = |V(G)|-1$$
, which answers a question posed by Ayyaswamy and Natarajan [An. Stt. Univ. Ovidius Constanta 23(2):187–199, 2015]. We present probabilistic upper bounds for the hop domination number. We also prove that almost all graphs $$G=G(n,p(n))$$
have a hop dominating set of cardinality at most the total domination number if $$p(n)\ll 1/n$$
, and almost all graphs $$G=G(n,p(n))$$
have a hop dominating set of cardinality at most $$1+np(1+o(1))$$
, if p is constant. We show that the decision problems for the 2-step dominating set and hop dominating set problems are NP-complete for planar bipartite graphs and planar chordal graphs.
TL;DR: This work proposed a distributed algorithm to find a connected dominating set for unit disk graphs and proposed a scheduling scheme that obtains O() conflict-free time slots to deal with interference.
TL;DR: This paper provides two lower bounds on the double ve-domination number of trees and unicycle graphs in terms of the order n, the number of leaves and support vertices, and it is proved that for every nontrivial connected graphs G, γdve(G) ≥ γve( G) + 1.
Abstract: A vertex v of a graph G = (V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (double vertex-edge dominating set, respectively) if every edge of E is ve-dominated by at least one vertex (at least two vertices) of S. The minimum cardinality of a vertex-edge dominating set (double vertex-edge dominating set, respectively) of G is the vertex-edge domination number γve(G) (the double vertex-edge domination number γdve(G), respectively). In this paper, we initiate the study of double vertex-edge domination. We first show that determining the number γdve(G) for bipartite graphs is NP-complete. We also prove that for every nontrivial connected graphs G, γdve(G) ≥ γve(G) + 1, and we characterize the trees T with γdve(T) = γve(T) + 1 or γdve(T) = γve(T) + 2. Finally, we provide two lower bounds on the double ve-domination number of trees and unicycle graphs in terms of the order n, the number of leaves and support vertices, and we characterize the trees attaining the lower bound.
TL;DR: This note presents an alternative proof of the result by Dorfling et al. establishing that any maximal outerplanar graph of order n≥5 has a total dominating set of size at most⌊2n5⌋, apart from two exceptions.
Abstract: In this note we present an alternative proof of the result by Dorfling et al. (Discrete Math 339(3):1180–1188, 2016) establishing that any maximal outerplanar graph of order \(n \ge 5\) has a total dominating set of size at most \(\lfloor \frac{2n}{5}\rfloor \), apart from two exceptions. In addition, we briefly discuss a relation between total domination in maximal outerplanar graphs and the concept of watched guards in simple polygons.
31 Jan 2017
TL;DR: This work provides efficient quadratic unconstrained binary optimization formulations for the Dominating Set and Edge Cover combinatorial problems suitable for adiabatic quantum computers, which are viewed as a real-world enhanced model of simulated annealing.
Abstract: We provide efficient quadratic unconstrained binary optimization (QUBO) formulations for the Dominating Set and Edge Cover combinatorial problems suitable for adiabatic quantum computers, which are viewed as a real-world enhanced model of simulated annealing (e.g. a type of genetic algorithm with quantum tunneling). The number of qubits (dimension of QUBO matrices) required to solve these set cover problems are O(n + n lg n) and O(m + n lg n) respectively, where n is the number of vertices and m is the number of edges. We also extend our formulations for the Minimum Vertex-Weighted Dominating Set problem and the Minimum Edge-Weighted Edge Cover problem. Experimental results for the Dominating Set and Edge Cover problems using a D-Wave Systems quantum computer with 1098 active qubit-coupled processors are also provided for a selection of known common graphs.
Proceedings Article•
01 Jan 2017TL;DR: In this article, the complexity of the (k, r ) -center problem with respect to various standard graph parameters was shown to be 2 Θ( td 2 ) for any r ≥ 1.
Abstract: In ( k , r ) - Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: • For any r ≥ 1 , we show an algorithm that solves the problem in O ∗ ( ( 3 r + 1 ) cw ) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm’s performance. As a corollary, for r = 1 , this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw . • We strengthen previously known FPT lower bounds, by showing that ( k , r ) - Center is W[1]-hard parameterized by the input graph’s vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs. • We show that the complexity of the problem parameterized by tree-depth is 2 Θ ( td 2 ) , by showing an algorithm of this complexity and a tight ETH-based lower bound. We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth, which work efficiently independently of the values of k , r . In particular, we give algorithms which, for any ϵ > 0 , run in time O ∗ ( ( tw ∕ ϵ ) O ( tw ) ) , O ∗ ( ( cw ∕ ϵ ) O ( cw ) ) and return a ( k , ( 1 + ϵ ) r ) -center if a ( k , r ) -center exists, thus circumventing the problem’s W-hardness.
TL;DR: A hybrid framework called collaborative topology control protocol, which combines dominating set-based clustering and transmission power adjustment, is proposed, which is capable of versatile performance in terms of transmission range/energy cost, the number of neighbors, edges, and hop distance.
Abstract: Topology control is relevant in wireless sensor network because of two reasons, namely minimal sensor coverage and power constraints. The former condition is typically satisfied by high-density deployment, whereas the latter mainly concerns with the control protocol design that is adaptable. Controlling communication topology is at the center of the efforts to optimize network performance while improving energy conservation. A dense topology often results in high interference and lower spatial reuse thus reduced capacity, while sparse topology is susceptible to network partitioning and sub-optimal path selection from the routing layer. Topology control has been extensively studied in both flat and hierarchical network by mean of power adjustment and clustering, respectively. Despite a common goal of making the topology less complex both techniques differ in their approach. While the focus of clustering is to form a connected backbone which consists of a minimum subset of nodes, i.e., dominating set, power adjustment focuses on minimizing energy consumption. Combining both approaches remains a relatively lesser explored area. We proposed a hybrid framework called collaborative topology control protocol, which combines dominating set-based clustering and transmission power adjustment. The protocol operates in two stages. During the first stage, a parameterized minimum virtual connected dominating set algorithm is executed to obtain clusters of various desirable properties. In the second stage, each cluster-head executes a distributed power adjustment algorithm. The simulation results show that the proposed topology control framework is capable of versatile performance in terms of transmission range/energy cost, the number of neighbors, edges, and hop distance. Moreover, the topology construction process uses the locally available information only with minimal communication overhead.
TL;DR: It is proved that the class of claw-free cubic graphs has a vertex set that can be partitioned into two total dominating sets, and that this property also holds for a connected, cubic graph that is chordal or 4 -chordal.
Posted Content•
TL;DR: Henning, Klav\v{z}ar, and Rall as mentioned in this paper showed that the sharp upper bound for the game total domination number is at least 11−14 n for graphs with no isolated vertices or isolated edges.
Abstract: The total domination game is a two-person competitive optimization game, where the players, Dominator and Staller, alternately select vertices of an isolate-free graph $G$. Each vertex chosen must strictly increase the number of vertices totally dominated. This process eventually produces a total dominating set of $G$. Dominator wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game total domination number of $G$, $\gamma_{{\rm tg}}(G)$, is the number of vertices chosen when Dominator starts the game and both players play optimally.
Recently, Henning, Klav\v{z}ar, and Rall proved that $\gamma_{{\rm tg}}(G) \le \frac{4}{5}n$ holds for every graph $G$ which is given on $n$ vertices such that every component of it is of order at least $3$; they also conjectured that the sharp upper bound would be $\frac{3}{4}n$. Here, we prove that $\gamma_{{\rm tg}}(G)\le \frac{11}{14}n$ holds for every $G$ which contains no isolated vertices or isolated edges.
20 Apr 2017
TL;DR: A new hybrid heuristic for the Dominating Set problem that further improves this method is obtained by combining the turbo-charging technique with other standard heuristic tools including Local Search (LS).
Abstract: Turbo-charging is a recent algorithmic technique that is based on the fixed-parameter tractability of the dynamic versions of some problems as a way to improve heuristics. We demonstrate the effectiveness of this technique and develop the turbo-charging idea further. A new hybrid heuristic for the Dominating Set problem that further improves this method is obtained by combining the turbo-charging technique with other standard heuristic tools including Local Search (LS). We implement both the recently proposed “turbo greedy” algorithm of Downey et al. [8] and a new method presented in this paper. The performance of these new heuristics is assessed on three suites of benchmark datasets, namely DIMACS, BHOSLIB and KONECT. Experiments comparing our algorithm to both heuristic and exact algorithms demonstrate its effectiveness. Our algorithm often produced results that were either exact or better than all the other available algorithms.
TL;DR: It is proved that if G is a graph that is both perfect and irredundant perfect, or if G belongs to certain classes of well-covered graphs, then d 0 ( G) = Γ ( G ) + 1 .
05 Aug 2017
TL;DR: The secure restrained convex dominating sets in the join of two graphs are characterized and some realization problems are given and some important results are given.
Abstract: L et be a connected simple graph. A restrained convex dominating set in a connected graph is a secure restrained convex dominating set, if for each element in there exists an element in such that and is a restrained convex dominating set. The secure restrained convex domination number of , denoted by , is the minimum cardinality of a secure restrained convex dominating set in . A secure restrained convex dominating set of cardinality will be called a - . In this paper, we give some realization problems will be given. In particular, we show that given positive integers and such that and there exists a connected graph with and . Further, we characterize the secure restrained convex dominating sets in the join of two graphs and give some important results.
TL;DR: The weighted DIM problem is solvable in polynomial time for P_8-free graphs, which is NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three.
Abstract: Let $$G=(V,E)$$G=(V,E) be a finite undirected graph. An edge set $$E' \subseteq E$$E?⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of $$E'$$E?. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is $${\mathbb {NP}}$$NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three and is solvable in linear time for $$P_7$$P7-free graphs. However, its complexity was open for $$P_k$$Pk-free graphs for any $$k \ge 8$$k?8; $$P_k$$Pk denotes the chordless path with k vertices and $$k-1$$k-1 edges. We show in this paper that the weighted DIM problem is solvable in polynomial time for $$P_8$$P8-free graphs.
TL;DR: The first performance guaranteed approximation algorithm for inline-formula-CDS in unit disk graph, which models homogeneous WSNs, is presented.
Abstract: To save energy and alleviate interference, connected dominating set (CDS) was proposed to serve as a virtual backbone of wireless sensor networks (WSNs). Because sensor nodes may fail due to accidental damages or energy depletion, it is desirable to construct a fault tolerant virtual backbone with high redundancy in both coverage and connectivity. This can be modeled as a $k$ -connected $m$ -fold dominating set (abbreviated as $(k,m)$ -CDS) problem. A node set $C\subseteq V(G)$ is a $(k,m)$ -CDS of graph $G$ if every node in $V(G)\backslash C$ is adjacent with at least $m$ nodes in $C$ and the subgraph of $G$ induced by $C$ is $k$ -connected. Constant approximation algorithm is known for $(3,m)$ -CDS in unit disk graph, which models homogeneous WSNs. In this paper, we present the first performance guaranteed approximation algorithm for $(3,m)$ -CDS in a heterogeneous WSN. In fact, our performance ratio is valid for any topology. The performance ratio is at most $\gamma $ , where $\gamma =\alpha +8+2\ln (2\alpha -6)$ for $\alpha \geq 4$ and $\gamma =3\alpha +2\ln 2$ for $\alpha , and $\alpha $ is the performance ratio for the minimum $(2,m)$ -CDS problem. Using currently best known value of $\alpha $ , the performance ratio is $\ln \delta +o(\ln \delta )$ , where $\delta $ is the maximum degree of the graph, which is asymptotically best possible in view of the non-approximability of the problem. Applying our algorithm on a unit disk graph, the performance ratio is less than 27, improving previous ratio 62.3 by a large amount for the $(3,m)$ -CDS problem on a unit disk graph.
TL;DR: This paper proposes a new polynomial time constant factor approximation algorithm for a more-a-decade-long open NP-hard problem, the minimum four-connected unit disk graph (UDG) with any positive integer, using LaTeX notation for the first time in the literature.
Abstract: This paper proposes a new polynomial time constant factor approximation algorithm for a more-a-decade-long open NP-hard problem, the minimum four-connected $m$ -dominating set problem in unit disk graph (UDG) with any positive integer $m \geq 1$ for the first time in the literature. We observe that it is difficult to modify the existing constant factor approximation algorithm for the minimum three-connected $m$ -dominating set problem to solve the minimum four-connected $m$ -dominating set problem in UDG due to the structural limitation of Tutte decomposition, which is the main graph theory tool used by Wang et al. to design their algorithm. To resolve this issue, we first reinvent a new constant factor approximation algorithm for the minimum three-connected $m$ -dominating set problem in UDG and later use this algorithm to design a new constant factor approximation algorithm for the minimum four-connected $m$ -dominating set problem in UDG.