Showing papers on "Dominating set published in 2015"

Domination in vague graphs and its applications

Abstract: The concept of vague graph introduced by Ramakrishna in (8). The main purpose of this paper is to introduce the concepts of cardinality, dominating set, independent set, total dominating number and independent dominating number of a vague graph. The notion of irredundance number of a vague graph is discussed, too. Finally we give an application of domination in vague graphs.

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Abstract: We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density.

DEADS: Depth and Energy Aware Dominating Set Based Algorithm for Cooperative Routing along with Sink Mobility in Underwater WSNs

Abstract: Performance enhancement of Underwater Wireless Sensor Networks (UWSNs) in terms of throughput maximization, energy conservation and Bit Error Rate (BER) minimization is a potential research area. However, limited available bandwidth, high propagation delay, highly dynamic network topology, and high error probability leads to performance degradation in these networks. In this regard, many cooperative communication protocols have been developed that either investigate the physical layer or the Medium Access Control (MAC) layer, however, the network layer is still unexplored. More specifically, cooperative routing has not yet been jointly considered with sink mobility. Therefore, this paper aims to enhance the network reliability and efficiency via dominating set based cooperative routing and sink mobility. The proposed work is validated via simulations which show relatively improved performance of our proposed work in terms the selected performance metrics.

Statistical Mechanics of the Minimum Dominating Set Problem

Abstract: The minimum dominating set (MDS) problem has wide applications in network science and related fields. It aims at constructing a node set of smallest size such that any node of the network is either in this set or is adjacent to at least one node of this set. Although this optimization problem is generally very difficult, we show it can be exactly solved by a generalized leaf-removal (GLR) process if the network contains no core. We present a percolation theory to describe the GLR process on random networks, and solve a spin glass model by mean field method to estimate the MDS size. We also implement a message-passing algorithm and a local heuristic algorithm that combines GLR with greedy node-removal to obtain near-optimal solutions for single random networks. Our algorithms also perform well on real-world network instances.

A PTAS for the Weighted Unit Disk Cover Problem

Abstract: We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.

Algorithmic Applications of Tree-Cut Width

Abstract: The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the potential to facilitate the efficient solution of problems which are not known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We introduce the notion of nice tree-cut decompositions and provide FPT algorithms for the showcase problems Capacitated Vertex Cover, Capacitated Dominating Set and Imbalance parameterized by the tree-cut width of an input graph G. On the other hand, we show that List Coloring, Precoloring Extension and Boolean CSP (the latter parameterized by the tree-cut width of the incidence graph) are W[1]-hard and hence unlikely to be fixed-parameter tractable when parameterized by tree-cut width.

Polynomial Delay Algorithm for Listing Minimal Edge Dominating Sets in Graphs

Abstract: It was proved independently and with different techniques in [Golovach et al. - ICALP 2013] and [Kante et al. - ISAAC 2012] that there exists an incremental output polynomial algorithm for the enumeration of the minimal edge dominating sets in graphs, i.e., minimal dominating sets in line graphs. We provide the first polynomial delay and polynomial space algorithm for the problem. We propose a new technique to enlarge the applicability of Berge’s algorithm that is based on skipping hard parts of the enumeration by introducing a new search strategy. The new search strategy is given by a strong use of the structure of line graphs.

An Incremental Polynomial Time Algorithm to Enumerate All Minimal Edge Dominating Sets

Abstract: We show that all minimal edge dominating sets of a graph can be generated in incremental polynomial time. We present an algorithm that solves the equivalent problem of enumerating minimal (vertex) dominating sets of line graphs in incremental polynomial, and consequently output polynomial, time. Enumeration of minimal dominating sets in graphs has recently been shown to be equivalent to enumeration of minimal transversals in hypergraphs. The question whether the minimal transversals of a hypergraph can be enumerated in output polynomial time is a fundamental and challenging question; it has been open for several decades and has triggered extensive research. To obtain our result, we present a flipping method to generate all minimal dominating sets of a graph. Its basic idea is to apply a flipping operation to a minimal dominating set $$D^*$$D? to generate minimal dominating sets $$D$$D such that $$G[D]$$G[D] contains more edges than $$G[D^*]$$G[D?]. We show that the flipping method works efficiently on line graphs, resulting in an algorithm with delay $$O(n^2m^2|\mathcal {L}|)$$O(n2m2|L|) between each pair of consecutively output minimal dominating sets, where $$n$$n and $$m$$m are the numbers of vertices and edges of the input graph, respectively, and $$\mathcal {L}$$L is the set of already generated minimal dominating sets. Furthermore, we are able to improve the delay to $$O(n^2m|\mathcal {L}|)$$O(n2m|L|) on line graphs of bipartite graphs. Finally we show that the flipping method is also efficient on graphs of large girth, resulting in an incremental polynomial time algorithm to enumerate the minimal dominating sets of graphs of girth at least 7.

An optimization algorithm for the minimum k-connected m-dominating set problem in wireless sensor networks

Abstract: In wireless sensor networks (WSNs), virtual backbone has been proposed as the routing infra-structure and connected dominating set has been widely adopted as virtual backbone. However, since the sensors in WSNs are prone to failures, recent studies suggest that it is also important to maintain a certain degree of redundancy in the backbone. To construct a robust backbone, so called k-connected m-dominating set has been proposed. In this research, we propose an integer programming formulation and an optimal algorithm for the minimum k-connected m-dominating set problem. To the best of our knowledge, this is the first integer programming formulation for the problem, and extensive computational results show that our optimal algorithm is capable of finding a solution within a reasonable amount of time.

A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs

Abstract: An output-polynomial algorithm for the listing of minimal dominating sets in graphs is a challenging open problem and is known to be equivalent to the well-known Transversal problem which asks for an output-polynomial algorithm for listing the set of minimal transversals in hypergraphs. We give a polynomial delay algorithm to list the set of minimal dominating sets in chordal graphs, an important and well-studied graph class where such an algorithm was not known. The algorithm uses a new decomposition method of chordal graphs based on clique trees.

An Integer Programming Approach for Fault-Tolerant Connected Dominating Sets

Abstract: This paper considers the minimum k-connected d-dominating set problem, which is a fault-tolerant generalization of the minimum connected dominating set (MCDS) problem. Three integer programming formulations based on vertex cuts are proposed (depending on whether d k) and their integer hulls are studied. The separation problem for the vertex-cut inequalities is a weighted vertex-connectivity problem and is polytime solvable, meaning that the LP relaxation can be solved in polytime despite having exponentially many constraints. A new class of valid inequalities—r-robust vertex-cut inequalities—is introduced and is shown to induce exponentially many facets. Finally, a lazy-constraint approach is shown to compare favorably with existing approaches for the MCDS problem (the case k = d = 1), and is in fact the fastest in literature for standard test instances. A key subroutine is an algorithm for finding an inclusion-wise minimal vertex cut in linear time. Computational results for (k, d) = (...

Subgraph Matching with Set Similarity in a Large Graph Database

Abstract: In real-world graphs such as social networks, Semantic Web and biological networks, each vertex usually contains rich information, which can be modeled by a set of tokens or elements. In this paper, we study a subgraph matching with set similarity (SMS $^2$ ) query over a large graph database, which retrieves subgraphs that are structurally isomorphic to the query graph, and meanwhile satisfy the condition of vertex pair matching with the (dynamic) weighted set similarity. To efficiently process the SMS $^2$ query, this paper designs a novel lattice-based index for data graph, and lightweight signatures for both query vertices and data vertices. Based on the index and signatures, we propose an efficient two-phase pruning strategy including set similarity pruning and structure-based pruning, which exploits the unique features of both (dynamic) weighted set similarity and graph topology. We also propose an efficient dominating-set-based subgraph matching algorithm guided by a dominating set selection algorithm to achieve better query performance. Extensive experiments on both real and synthetic datasets demonstrate that our method outperforms state-of-the-art methods by an order of magnitude.

The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation

Abstract: A minimum dominating set for a digraph (directed graph) is a smallest set of vertices such that each vertex either belongs to this set or has at least one parent vertex in this set. We solve this hard combinatorial optimization problem approximately by a local algorithm of generalized leaf removal and by a message-passing algorithm of belief propagation. These algorithms can construct near-optimal dominating sets or even exact minimum dominating sets for random digraphs and also for real-world digraph instances. We further develop a core percolation theory and a replica-symmetric spin glass theory for this problem. Our algorithmic and theoretical results may facilitate applications of dominating sets to various network problems involving directed interactions.

Independence and Efficient Domination on $P_6$-free Graphs

Abstract: In the Independent set problem, the input is a graph $G$, every vertex has a non-negative integer weight, and the task is to find a set $S$ of pairwise non-adjacent vertices, maximizing the total weight of the vertices in $S$. We give an $n^{O (\log^2 n)}$ time algorithm for this problem on graphs excluding the path $P_6$ on $6$ vertices as an induced subgraph. Currently, there is no constant $k$ known for which Independent Set on $P_{k}$-free graphs becomes NP-complete, and our result implies that if such a $k$ exists, then $k > 6$ unless all problems in NP can be decided in (quasi)polynomial time. Using the combinatorial tools that we develop for the above algorithm, we also give a polynomial-time algorithm for Efficient Dominating Set on $P_6$-free graphs. In this problem, the input is a graph $G$, every vertex has an integer weight, and the objective is to find a set $S$ of maximum weight such that every vertex in $G$ has exactly one vertex in $S$ in its closed neighborhood, or to determine that no such set exists. Prior to our work, the class of $P_6$-free graphs was the only class of graphs defined by a single forbidden induced subgraph on which the computational complexity of Efficient Dominating Set was unknown.

Hop Domination in Graphs-II

Abstract: Let G = (V;E) be a graph. A set S V (G) is a hop dominating set of G if for every v 2 V S, there exists u 2 S such that d(u;v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by h(G). In this paper we characterize the family of trees and unicyclic graphs for which h(G) = t(G) and h(G) = c(G) where t(G) and c(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

Approximation Algorithm for Minimum Weight (k,m)-CDS Problem in Unit Disk Graph

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 work, 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 $(\alpha+2.5k\rho)$ 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.)

Bounds on the hop domination number of a tree

Abstract: A hop dominating set of a graph G is a set D of vertices of G if for every vertex of V(G)∖D, there exists u∈D such that d(u,v)=2. The hop domination number of a graph G, denoted by γ h (G), is the minimum cardinality of a hop dominating set of G. We prove that for every tree T of order n with l leaves and s support vertices we have (n−l−s+4)/3≤γ h (G)≤n/2, and characterize the trees attaining each of the bounds.