Showing papers on "Dominating set published in 2012"

Triple Connected Domination Number of a Graph

Abstract: The concept of triple connected graphs with real life application was introduced in (7) by considering the existence of a path containing any three vertices of a graph G. In this paper, we introduce a new domination parameter, called Smarandachely triple connected domination number of a graph. A subset S of V of a nontrivial graph G is said to be Smarandachely triple connected dominating set, if S is a dominating set and the induced sub graph h Si is triple connected. The minimum cardinality taken over all Smarandachely triple connected dominating sets is called the Smarandachely triple connected domination number and is denoted by tc. We determine this number for some standard graphs and obtain bounds for general graphs. Its relationship with other graph theoretical parameters are also investigated.

Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling

Abstract: The minimum-weight set cover problem is widely known to be O(log n)-approximable, with no improvement possible in the general case. We take the approach of exploiting problem structure to achieve better results, by providing a geometry-inspired algorithm whose approximation guarantee depends solely on an instance-specific combinatorial property known as shallow cell complexity (SCC). Roughly speaking, a set cover instance has low SCC if any column-induced submatrix of the corresponding element-set incidence matrix has few distinct rows. By adapting and improving Varadarajan's recent quasi-uniform random sampling method for weighted geometric covering problems, we obtain strong approximation algorithms for a structurally rich class of weighted covering problems with low SCC. We also show how to derandomize our algorithm.Our main result has several immediate consequences. Among them, we settle an open question of Chakrabarty et al. [8] by showing that weighted instances of the capacitated covering problem with underlying network structure have O(1)-approximations. Additionally, our improvements to Varadarajan's sampling framework yield several new results for weighted geometric set cover, hitting set, and dominating set problems. In particular, for weighted covering problems exhibiting linear (or near-linear) union complexity, we obtain approximability results agreeing with those known for the unweighted case. For example, we obtain a constant approximation for the weighted disk cover problem, improving upon the 2O(log* n)-approximation known prior to our work and matching the O(1)-approximation known for the unweighted variant.

Weak compositions and their applications to polynomial lower bounds for kernelization

Abstract: In this paper we use the notion of weak compositions to obtain polynomial kernelization lower-bounds for several natural parameterized problems Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1}* x N be two parameterized problems where the unparameterized version of L1 is NP-hard Assuming coNP n NP/poly, our framework essentially states that composing tL1-instances each with parameter k, to an L2-instance with parameter k' ≤ t1/dkO(1), implies that L2 does not have a kernel of size O(kd−e) for any e > 0 We show two examples of weak composition and derive polynomial kernelization lower bounds for d-Bipartite Regular Perfect Code and d-Dimensional Matching, parameterized by the solution size k By reduction, using linear parameter transformations, we then derive the following lower-bounds for kernel sizes when the parameter is the solution size k (assuming coNP n NP/poly):• d-Set Packing, d-Set Cover, d-Exact Set Cover, Hitting Set with d-Bounded Occurrences, and Exact Hitting Set with d-Bounded Occurrences have no kernels of size O(kd−3−e) for any e > 0• Kd Packing and Induced K1,d Packing have no kernels of size O(kd−4−e) for any e > 0• d-Red-Blue Dominating Set and d-Steiner Tree have no kernels of sizes O(kd−3−e) and O(kd−4−e), respectively, for any e > 0Our results give a negative answer to an open question raised by Dom, Lokshtanov, and Saurabh [ICALP2009] regarding the existence of uniform polynomial kernels for the problems above All our lower bounds transfer automatically to compression lower bounds, a notion defined by Harnik and Naor [SICOMP2010] to study the compressibility of NP instances with cryptographic applications We believe weak composition can be used to obtain polynomial kernelization lower bounds for other interesting parameterized problemsIn the last part of the paper we strengthen previously known super-polynomial kernelization lower bounds to super-quasi-polynomial lower bounds, by showing that quasi-polynomial kernels for compositional NP-hard parameterized problems implies the collapse of the exponential hierarchy These bounds hold even the kernelization algorithms are allowed to run in quasipolynomial time

On Problems as Hard as CNF-SAT

Abstract: The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and 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(2^n), 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(2^n), 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 epsilon

Dominating Sets and Domination Polynomials of Graphs

Abstract: Let be a simple graph. A set is a dominating set of , if every vertex in is adjacent to at least one vertex in . Let be the family of all dominating sets of a path with cardinality , and let . In this paper, we construct , and obtain a recursive formula for . Using this recursive formula, we consider the polynomial , which we call domination polynomial of paths and obtain some properties of this polynomial.

Generating pictorial storylines via minimum-weight connected dominating set approximation in multi-view graphs

Abstract: This paper introduces a novel framework for generating pictorial storylines for given topics from text and image data on the Internet. Unlike traditional text summarization and timeline generation systems, the proposed framework combines text and image analysis and delivers a storyline containing textual, pictorial, and structural information to provide a sketch of the topic evolution. A key idea in the framework is the use of an approximate solution for the dominating set problem. Given a collection of topic-related objects consisting of images and their text descriptions, a weighted multi-view graph is first constructed to capture the contextual and temporal relationships among these objects. Then the objects are selected by solving the minimum-weighted connected dominating set problem defined on this graph. Comprehensive experiments on real-world data sets demonstrate the effectiveness of the proposed framework.

Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

Abstract: We show that for every fixed j ≥ i ≥ 1, the k-Dominating Set problem restricted to graphs that do not have Kij (the complete bipartite graph on (i + j) vertices, where the two parts have i and j vertices, respectively) as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. We describe a polynomial-time algorithm that, given a Ki,j-free graph G and a nonnegative integer k, constructs a graph H (the “kernel”) and an integer k' such that (1) G has a dominating set of size at most k if and only if H has a dominating set of size at most k', (2) H has O((j + 1)i + 1ki2) vertices, and (3) k' = O((j + 1)i + 1ki2).Since d-degenerate graphs do not have Kd+1,d+1 as a subgraph, this immediately yields a polynomial kernel on O((d + 2)d+2k(d + 1)2) vertices for the k-Dominating Set problem on d-degenerate graphs, solving an open problem posed by Alon and Gutner [Alon and Gutner 2008; Gutner 2009].The most general class of graphs for which a polynomial kernel was previously known for k-Dominating Set is the class of Kh-topological-minor-free graphs [Gutner 2009]. Graphs of bounded degeneracy are the most general class of graphs for which an FPT algorithm was previously known for this problem. Kh-topological-minor-free graphs are Ki,j-free for suitable values of i,j (but not vice-versa), and so our results show that k-Dominating Set has both FPT algorithms and polynomial kernels in strictly more general classes of graphs.Using the same techniques, we also obtain an O(jki) vertex-kernel for the k-Independent Dominating Set problem on Ki,j-free graphs.

Solving the Connected Dominating Set Problem and Power Dominating Set Problem by Integer Programming

Abstract: In this paper, we propose several integer programming approaches with a polynomial number of constraints to formulate and solve the minimum connected dominating set problem. Further, we consider both the power dominating set problem – a special dominating set problem for sensor placement in power systems – and its connected version. We propose formulations and algorithms to solve these integer programs, and report results for several power system graphs.

Linear kernels for (connected) dominating set on H-minor-free graphs

Abstract: We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given H-minor free graph G and positive integer k, output an H-minor free graph G' on O(k) vertices such that G has a (connected) dominating set of size k if and only if G' has. Prior to our work, the only polynomial kernel for Dominating Set on graphs excluding a fixed graph H as a minor was due to Alon and Gutner [ECCC 2008, IWPEC 2009] and to Philip, Raman, and Sikdar [ESA 2009] but the size of their kernel is kc(H), where c(H) is a constant depending on the size of H. Alon and Gutner asked explicitly, whether one can obtain a linear kernel for Dominating Set on H-minor free graphs. We answer this question in affirmative. For Connected Dominating Set no polynomial kernel on H-minor free graphs was known prior to our work.Our results are based on a novel generic reduction rule producing an equivalent instance of the problem with treewidth O(√k). The application of this rule in a divide-and-conquer fashion together with protrusion techniques brings us to linear kernels.As a byproduct of our results we obtain the first subexponential time algorithms for Connected Dominating Set, a deterministic algorithm solving the problem on an n-vertex H-minor free graph in time 2O(√k log k) + nO(1) and a Monte Carlo algorithm of running time 2O(√k) + nO(1). For Dominating Set our results implies a significant simplification and refinement of a 2O(√k)nO(1) algorithm on H minor free graphs due to Demaine et al. [SODA 2003, J. ACM 2005].

On the Independent Domination Number of Regular Graphs

Abstract: A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs.

Exact Algorithms for Edge Domination

Abstract: An edge dominating set in a graph G=(V,E) is a subset of the edges D⊆E such that every edge in E is adjacent or equal to some edge in D. The problem of finding an edge dominating set of minimum cardinality is NP-hard. We present a faster exact exponential time algorithm for this problem. Our algorithm uses O(1.3226n) time and polynomial space. The algorithm combines an enumeration approach of minimal vertex covers in the input graph with the branch and reduce paradigm. Its time bound is obtained using the measure and conquer technique. The algorithm is obtained by starting with a slower algorithm which is refined stepwisely. In each of these refinement steps, the worst cases in the measure and conquer analysis of the current algorithm are reconsidered and a new branching strategy is proposed on one of these worst cases. In this way a series of algorithms appears, each one slightly faster than the previous one, ending in the O(1.3226n) time algorithm. For each algorithm in the series, we also give a lower bound on its running time. We also show that the related problems: minimum weight edge dominating set, minimum maximal matching and minimum weight maximal matching can be solved in O(1.3226n) time and polynomial space using modifications of the algorithm for edge dominating set. In addition, we consider the matrix dominating set problem which we solve in O(1.3226n+m) time and polynomial space for n×m matrices, and the parametrised minimum weight maximal matching problem for which we obtain an Oź(2.4179k) time and space algorithm.

On the Parameterized and Approximation Hardness of Metric Dimension

Abstract: The NP-hard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u,w} if the distance (length of a shortest path) between v and u is different from the distance of v and w. We give a polynomial-time computable reduction from the Bipartite Dominating Set problem to Metric Dimension on maximum degree three graphs such that there is a one-to-one correspondence between the solution sets of both problems. There are two main consequences of this: First, it proves that Metric Dimension on maximum degree three graphs is W[2]-complete with respect to the parameter k. This answers an open question concerning the parameterized complexity of Metric Dimension posed by D\'iaz et al. [ESA'12] and already mentioned by Lokshtanov [Dagstuhl seminar, 2009]. Additionally, it implies that Metric Dimension cannot be solved in n^{o(k)} time, unless the assumption FPT eq W[1] fails. This proves that a trivial n^{O(k)} algorithm is probably asymptotically optimal. Second, as Bipartite Dominating Set is inapproximable within o(log n), it follows that Metric Dimension on maximum degree three graphs is also inapproximable by a factor of o(log n), unless NP=P. This strengthens the result of Hauptmann et al. [JDA 2012] who proved APX-hardness on bounded-degree graphs.

k-tuple total domination in cross products of graphs

Abstract: For k?1 an integer, a set S of vertices in a graph G with minimum degree at least k is a k-tuple total dominating set of G if every vertex of G is adjacent to at least k vertices in S. The minimum cardinality of a k-tuple total dominating set of G is the k-tuple total domination number of G. When k=1, the k-tuple total domination number is the well-studied total domination number. In this paper, we establish upper and lower bounds on the k-tuple total domination number of the cross product graph G×H for any two graphs G and H with minimum degree at least k. In particular, we determine the exact value of the k-tuple total domination number of the cross product of two complete graphs.

Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs

Abstract: Let G = (V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v ∈ V dominates its closed neighborhood N[v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v ∈ V, there is exactly one d ∈ D dominating v. An edge set M ⊆ E is an efficient edge dominating (e.e.d.) set for G if it is an efficient dominating set in the line graph L(G) of G. The ED problem (EED problem, respectively) asks for the existence of an e.d. set (e.e.d. set, respectively) in the given graph.

Positive influence dominating sets in power-law graphs

Abstract: Finding the minimum Positive Influence Dominating Set (PIDS) is a problem arisen from the social network applications. The problem has been studied on general random graphs. However, the social networks is presented more precisely by power-law graphs. One of the most important properties of social networks is the power-law degree distribution. In this paper, we focus on the PIDS problem in power-law graphs and prove that the greedy algorithm has a constant approximation ratio. Simulation results also demonstrate that greedy algorithm can effectively select a small scale PIDS set.

Minimal dominating sets in graph classes: combinatorial bounds and enumeration

Abstract: The maximum number of minimal dominating sets that a graph on n vertices can have is known to be at most 1.7159n . This upper bound might not be tight, since no examples of graphs with 1.5705n or more minimal dominating sets are known. For several classes of graphs, we substantially improve the upper bound on the maximum number of minimal dominating sets in graphs on n vertices. In some cases, we provide examples of graphs whose number of minimal dominating sets exactly matches the proved upper bound for that class, thereby showing that these bounds are tight. For all considered graph classes, the upper bound proofs are constructive and can easily be transformed into algorithms enumerating all minimal dominating sets of the input graph.

Independent domination on tree convex bipartite graphs

Abstract: An independent dominating set in a graph is a subset of vertices, such that every vertex outside this subset has a neighbor in this subset (dominating), and the induced subgraph of this subset contains no edge (independent). It was known that finding the minimum independent dominating set (Independent Domination) is $\cal{NP}$ -complete on bipartite graphs, but tractable on convex bipartite graphs. A bipartite graph is called tree convex, if there is a tree defined on one part of the vertices, such that for every vertex in another part, the neighborhood of this vertex is a connected subtree. A convex bipartite graph is just a tree convex one where the tree is a path. We find that the sum of larger-than-two degrees of the tree is a key quantity to classify the computational complexity of independent domination on tree convex bipartite graphs. That is, when the sum is bounded by a constant, the problem is tractable, but when the sum is unbounded, and even when the maximum degree of the tree is bounded, the problem is $\cal{NP}$ -complete.

Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time

Abstract: It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in terms of the weights). We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms, also for weighted and counting versions. For example, in this time we can solve the traveling salesman problem or count the number of Hamiltonian cycles. The rank-based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the matrix tree theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.

Independent transversal domination in graphs

Abstract: A set S ⊆ V of vertices in a graph G = (V,E) is called a dominating set if every vertex in V −S is adjacent to a vertex in S. A dominating set which intersects every maximum independent set in G is called an independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of G and is denoted by it(G). In this paper we begin an investigation of this parameter.