Minimum Dominating Set Problem Research Articles

A 4n-move self-stabilizing algorithm for the minimal dominating set problem using an unfair distributed daemon

A distributed system is self-stabilizing if, regardless of its initial state, the system is guaranteed to reach a legitimate (i.e., correct) state in finite time. In 2007, Turau proposed the first linear-time self-stabilizing algorithm for the minimal dominating set (MDS) problem under an unfair distributed daemon [9]; this algorithm stabilizes in at most 9n moves, where n is the number of nodes in the system. In 2008, Goddard et al. [4] proposed a 5n-move algorithm. In this paper, we present a 4n-move algorithm.

Efficient sub-5 approximations for minimum dominating sets in unit disk graphs

A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their applicability in wireless ad-hoc networks. Because the minimum dominating set problem for unit disk graphs is NP-hard, numerous approximation algorithms have been proposed in the literature, including some PTASs. However, since the proposal of a linear-time 5-approximation algorithm in 1995, the lack of efficient algorithms attaining better approximation factors has aroused attention. We introduce an O(n+m) algorithm that takes the usual adjacency representation of the graph as input and outputs a 44/9-approximation. This approximation factor is also attained by a second algorithm, which takes the geometric representation of the graph as input and runs in O(nlogn) time regardless of the number of edges. Additionally, we propose a 43/9-approximation which can be obtained in O(n2m) time given only the graph's adjacency representation. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.

A strengthened analysis of a local algorithm for the minimum dominating set problem in planar graphs

In recent years growing interest in local distributed algorithms has widely been observed. This results from their high resistance to errors and damage, as well as from their good performance, which is independent of the size of the network. A local deterministic distributed algorithm finding an approximation of a Minimum Dominating Set in planar graphs has been presented by Lenzen et al., and they proved that the algorithm returns a 130-approximation of the Minimum Dominating Set. In this article we will show that the algorithm is two times more effective than was previously assumed, and we prove that the algorithm by Lenzen et al. outputs a 52-approximation to a Minimum Dominating Set. Therefore the gap between the lower bound and the approximation ratio of the best yet local deterministic distributed algorithm is reduced by half.

Approximating 2-cliques in unit disk graphs

This paper studies distance-based clique relaxations in unit disk graphs arising in wireless networking applications. Namely, a 2-clique is a subset of nodes with pairwise distance at most two in the graph, and a 2-club is a subset of nodes inducing a subgraph of diameter two. It is shown that in a unit disk graph any 2-clique is 4-dominated and any 2-club is 3-dominated. The former observation is used to develop a 12-approximation algorithm for the maximum 2-clique problem in unit disk graphs. Moreover, this also implies polynomial solvability of the minimum dominating set problem in unit disk graphs of diameter two, whereas the same problem is shown to be hard in general diameter-two graphs. The paper also poses several related open questions of interest.

Graph Neural Network for Minimum Dominating Set

The dominating set concept in graphs has been used in many applications. In large graphs finding the minimum dominating set is difficult. The minimum dominating set problem in graphs seek a set D of minimum number of vertices such that each vertex of the graph is either in D or adjacent to a vertex in D. In a graph on n nodes if there is a single node of degree n-1 then that single node forms a minimum dominating set. In the proposed work, a designed network called Graph Neural Network (GNN) is used to identify the node of degree N-1 in a graph having a single node of degree N-1 as it forms the minimum dominating set in the graph.. The network is simulated for graphs with nodes varying from 5 to 15. The state dimension of the input vectors are analyzed for better convergence. It has been found that the minimum dominating set was correctly identified from 80% to 90% of the graphs when the state dimension was 2 and 3. It has been observed that when the state dimension was 2, the convergence was fast as it requires minimum hidden neurons than other state dimensions. It has also been observed that when the state dimension was greater than 3, convergence requires hours of time and more number of hidden neurons. GNN was able to identify the minimum dominating set in a graph on n vertices which has a single node of degree n-1.

Adaptive Approximate Data Collection for Wireless Sensor Networks

Data collection is a fundamental task in wireless sensor networks. In many applications of wireless sensor networks, approximate data collection is a wise choice due to the constraints in communication bandwidth and energy budget. In this paper, we focus on efficient approximate data collection with prespecified error bounds in wireless sensor networks. The key idea of our data collection approach ADC (Approximate Data Collection) is to divide a sensor network into clusters, discover local data correlations on each cluster head, and perform global approximate data collection on the sink node according to model parameters uploaded by cluster heads. Specifically, we propose a local estimation model to approximate the readings of sensor nodes in subsets, and prove rated error-bounds of data collection using this model. In the process of model-based data collection, we formulate the problem of selecting the minimum subset of sensor nodes into a minimum dominating set problem which is known to be NP-hard, and propose a greedy heuristic algorithm to find an approximate solution. We further propose a monitoring algorithm to adaptively adjust the composition of node subsets according to changes of sensor readings. Our trace-driven simulations demonstrate that ADC remarkably reduces communication cost of data collection with guaranteed error bounds.

Improved approximation for spanning star forest in dense graphs

A spanning subgraph of a graph G is called a spanning star forest of G if it is a collection of node-disjoint trees of depth at most 1. The size of a spanning star forest is the number of leaves in all its components. The goal of the spanning star forest problem is to find the maximum-size spanning star forest of a given graph. In this paper, we study the spanning star forest problem on c-dense graphs, where for any fixed c?(0,1), a graph of n vertices is called c-dense if it contains at least cn 2/2 edges. We design a $(\alpha+(1-\alpha)\sqrt{c}-\epsilon)$ -approximation algorithm for spanning star forest in c-dense graphs for any ?>0, where $\alpha=\frac{193}{240}$ is the best known approximation ratio of the spanning star forest problem in general graphs. Thus, our approximation ratio outperforms the best known bound for this problem when dealing with c-dense graphs. We also prove that, for any constant c?(0,1), approximating spanning star forest in c-dense graphs is APX-hard. We then demonstrate that for weighted versions (both node- and edge-weighted) of this problem, we cannot get any approximation algorithm with strictly better performance guarantee on c-dense graphs than on general graphs. Finally, we give strong inapproximability results for a closely related problem, namely the minimum dominating set problem, restricted on c-dense graphs.

Simulated annealing with stochastic local search for minimum dominating set problem

In this paper, we propose a new method based on simulated annealing (SA) to solve the minimum dominating set problem. To deal with the considered problem, a stochastic local search (SLS) method is built first to find local solutions next to given solutions. Then, a simulated annealing algorithm is invoked to enhance the SLS method with the ability of escaping from local solutions. Moreover, three trial solution generation mechanisms are used to improve iterate solutions. The experimental results have shown the promising performance of the proposed SA-based method in comparison with some other meta-heuristics in terms of solution qualities and computational costs.

Simulated Annealing 알고리즘을 이용한 최소 Dominating Set 문제의 효율성 증가에 대한 연구

The minimum dominating set problem of a graph G is to find a smallest possible dominating set. The minimum dominating set problem is a well-known NP-complete problem such that it cannot be solved in polynomial time. Heuristic or approximation algorithm, however, will perform well in certain area of application. In this paper, we suggest three different simulated annealing algorithms and experimentally show better efficiency improvement by applying these algorithms to the graph instances developed by DIMACS.

Theory and application of width bounded geometric separators

We introduce the notion of the width bounded geometric separator and develop the techniques for the existence of the width bounded separator in any fixed d-dimensional Euclidean space. The separator is applied in obtaining 2 O ( n ) time exact algorithms for a class of NP-complete geometric problems, whose previous algorithms take n O ( n ) time. One of those problems is the well-known disk covering problem, which seeks to determine the minimal number of fixed size disks to cover n points on a plane. They also include some NP-hard problems on disk graphs such as the maximum independent set problem, the vertex cover problem, and the minimum dominating set problem. For a constant a > 0 and a set of points Q on the plane, an a-wide separator is the region between two parallel lines of distance a that partitions Q into Q 1 (on the left side of the region), S (inside the region), and Q 2 (on the right side of the region). If the distance is at least one between every two points in the set Q with n points, there is an a-wide separator that partitions Q into Q 1 , S and Q 2 such that | Q 1 | , | Q 2 | ⩽ ( 2 / 3 ) n , and | S | ⩽ 1.2126 a n . Our width bounded separator in d-dimensional Euclidean space with fixed d is controlled by two parallel hyper-planes, and is used to design 2 O ( n 1 − 1 d ) -time algorithms for the d-dimensional disk covering problem and the above other problems in the d-dimensional disk graphs.

Exponential time algorithms for the minimum dominating set problem on some graph classes

The minimum dominating set problem remains NP-hard when restricted to any of the following graph classes: c -dense graphs, chordal graphs, 4-chordal graphs, weakly chordal graphs, and circle graphs. Developing and using a general approach, for each of these graph classes we present an exponential time algorithm solving the minimum dominating set problem faster than the best known algorithm for general graphs. Our algorithms have the following running time: O (1.4124 n ) for chordal graphs, O (1.4776 n ) for weakly chordal graphs, O (1.4845 n ) for 4-chordal graphs, O (1.4887 n ) for circle graphs, and O (1.2273 (1+√1−2 c ) n ) for c -dense graphs.

Formula omitted]-join decomposable graphs and algorithms with runtime single exponential in rankwidth

We introduce H -join decompositions of graphs, indexed by a fixed bipartite graph H . These decompositions are based on a graph operation that we call a H -join, which adds edges between two given graphs by taking partitions of their two vertex sets, identifying the classes of the partitions with vertices of H , and connecting classes by the pattern H . H -join decompositions are related to modular, split and rank decompositions. Given an H -join decomposition of an n -vertex m -edge graph G , we solve the Maximum Independent Set and Minimum Dominating Set problems on G in time O ( n ( m + 2 O ( ρ ( H ) 2 ) ) ) , and the q -Coloring problem in time O ( n ( m + 2 O ( q ρ ( H ) 2 ) ) ) , where ρ ( H ) is the rank of the adjacency matrix of H over GF(2). Rankwidth is a graph parameter introduced by Oum and Seymour, based on ranks of adjacency matrices over GF(2). For any positive integer k we define a bipartite graph R k and show that the graphs of rankwidth at most k are exactly the graphs having an R k -join decomposition, thereby giving an alternative graph-theoretic definition of rankwidth that does not use linear algebra. Combining our results we get algorithms that, for a graph G of rankwidth k given with its width k rank-decomposition, solves the Maximum Independent Set problem in time O ( n ( m + 2 1 2 k 2 + 9 2 k × k 2 ) ) , the Minimum Dominating Set problem in time O ( n ( m + 2 3 4 k 2 + 23 4 k × k 3 ) ) and the q -Coloring problem in time O ( n ( m + 2 q 2 k 2 + 5 q + 4 2 k × k 2 q × q ) ) . These are the first algorithms for NP-hard problems whose runtimes are single exponential in the rankwidth. 1 1 For a polynomial function p o l y we call 2 p o l y ( k ) single exponential in k .

A self-stabilizing algorithm for constructing weakly connected minimal dominating sets

This paper presents a new distributed self-stabilizing algorithm for the weakly connected minimal dominating set problem. It assumes a self-stabilizing algorithm to compute a breadth-first tree. Using an unfair distributed scheduler the algorithm stabilizes in at most O ( n m A ) moves, where A is the number of moves to construct a breadth-first tree. All previously known algorithms required an exponential number of moves.

The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number

In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W[1]-hard under this parameterization? We do two things: We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G)≤k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O *(f(k)). The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.

A linear‐time algorithm for broadcast domination in a tree

AbstractThe broadcast domination problem is a variant of the classical minimum dominating set problem in which a transmitter of power p at vertex v is capable of dominating (broadcasting to) all vertices within distance p from v. Our goal is to assign a broadcast power f(v) to every vertex v in a graph such that ΣvεVf(v) is minimized, and such that every vertex u with f(u) = 0 is within distance f(v) of some vertex v with f(v)> 0. The problem is solvable in polynomial time on a general graph (Heggernes and Lokshtanov, Disc Math (2006), 3267–3280) and Blair et al. (Congr. Num. (2004), 55–77.) gave an O(n2) algorithm for trees. In this article, we provide an O(n) algorithm for trees. Our algorithm is notable due to the fact that it makes decisions for each vertex v based on “nonlocal” information from vertices far away from v, whereas almost all other linear‐time algorithms for trees only make use of local information. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

Semi-nice tree-decompositions: The best of branchwidth, treewidth and pathwidth with one algorithm

Branchwidth and treewidth are connectivity parameters of graphs of high importance in algorithm design. By dynamic programming along the associated branch- or tree-decomposition one can solve most graph optimization problems in time linear in the graph size and exponential in the parameter. If one of these parameters is bounded on a class of graphs, then so is the other, but they can differ by a small constant factor and this difference can be crucial for the resulting runtime. In this paper we introduce semi-nice tree-decompositions and show that they combine the best of both branchwidth and treewidth. We first give simple algorithms to transform a given tree-decomposition or branch-decomposition into a semi-nice tree-decomposition. We then give two templates for dynamic programming along a semi-nice tree-decomposition, one for optimization problems over vertex subsets and another for optimization problems over edge subsets. We show that the resulting runtime will match or beat the runtimes achieved by doing dynamic programming directly on either a branch- or tree-decomposition. For example, given a graph G on n vertices with path-, tree- and branch-decompositions of width p w , t w and b w respectively, the Minimum Dominating Set problem on G is solved in time O ( n 2 min { 1.58 p w , 2 t w , 2.38 b w } ) by a single dynamic programming algorithm along a semi-nice tree-decomposition. On the applied side the immediate gain is that for each optimization problem one can achieve the benefits of both treewidth, branchwidth and pathwidth while developing and implementing only one dynamic programming algorithm. On the theoretical side the combination of the best properties of both branchwidth and treewidth in a single decomposition is a step towards a more general framework giving the fastest possible algorithms on tree-like graphs.

Approximation hardness of dominating set problems in bounded degree graphs

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, these bounds almost match the known upper bounds. The results are applied to improve the lower bounds for other related problems such as Maximum Induced Matching and Maximum Leaf Spanning Tree.

AN ENHANCED ANT COLONY OPTIMIZATION METAHEURISTIC FOR THE MINIMUM DOMINATING SET PROBLEM

This paper proposes an enhanced Ant Colony Optimization (ACO) metaheuristic called ACO-TS to attack the minimum dominating set (MDS) problem. One of the recognized difficulties faced by ACO in its original form is premature convergence, which produces less satisfactory solutions. We propose a way to encourage a higher degree of exploration of the search space by incorporating a technique based on a concept borrowed from genetic algorithms called tournament selection. Instead of always following the standard mechanism for selecting the next solution component, an ant would make its decision based on the outcome of a tournament between randomly selected allowable components. The frequency of the tournament selection is controlled by a probability measure. The use of tournament selection is coupled with an iteration-best pheromone update. To evaluate the enhanced ACO, we consider the MDS problem formulated from ad hoc network clustering. A comparison with its original form shows that the enhanced ACO produces better solutions using fewer number of cycles. We also empirically demonstrate that the proposed ACO produces better solutions than a genetic algorithm. Finally, we argue, based on empirical results, why the tournament selection approach is preferable to a pure random selection method.

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S ⊆ V , such that, for each u ∈ V , | N [ u ] ∩ S | ⩾ | N [ u ] | / 2 in a given graph G = ( V , E ) . We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs ( k ⩾ 5 ) , unless P = NP . We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2 k -regular graphs and to minimum monopoly problem for ( 2 k - 1 ) -regular graphs, where k ⩾ 3 . We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.

APX-hardness of domination problems in circle graphs

We show that the problem of finding a minimum dominating set in a circle graph is APX-hard: there is a constant δ > 0 such that there is no ( 1 + δ ) -approximation algorithm for the minimum dominating set problem on circle graphs unless P = NP . Hence a PTAS for this problem seems unlikely. This hardness result complements the ( 2 + ɛ ) -approximation algorithm for the problem [M. Damian, S.V. Pemmaraju, A ( 2 + ɛ ) -approximation scheme for minimum domination on circle graphs, J. Algorithms 42 (2) (2002) 255–276].