Showing papers on "Dominating set published in 2004"

An extended localized algorithm for connected dominating set formation in ad hoc wireless networks

Abstract: Efficient routing among a set of mobile hosts is one of the most important functions in ad hoc wireless networks. Routing based on a connected dominating set is a promising approach, where the search space for a route is reduced to the hosts in the set. A set is dominating if all the hosts in the system are either in the set or neighbors of hosts in the set. The efficiency of dominating-set-based routing mainly depends on the overhead introduced in the formation of the dominating set and the size of the dominating set. In this paper, we first review a localized formation of a connected dominating set called marking process and dominating-set-based routing. Then, we propose a dominant pruning rule to reduce the size of the dominating set. This dominant pruning rule (called Rule k) is a generalization of two existing rules (called Rule 1 and Rule 2, respectively). We prove that the vertex set derived by applying Rule k is still a connected dominating set. Rule k is more effective in reducing the dominating set derived from the marking process than the combination of Rules 1 and 2 and, surprisingly, in a restricted implementation with local neighborhood information, Rule k has the same communication complexity and less computation complexity. Simulation results confirm that Rule k outperforms Rules 1 and 2, especially in networks with relatively high vertex degree and high percentage of unidirectional links. We also prove that an upper bound exists on the average size of the dominating set derived from Rule k in its restricted implementation.

What cannot be computed locally

Abstract: We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors Ω(nc/k2/k) and Ω(Δ>1/k/k) for some constant c, where n and Δ denote the number of nodes and the largest degree in the graph. The number of rounds required in order to achieve a constant or even only a polylogarithmic approximation ratio is at least Ω(√log n/log log n) and Ω(logΔ/ log log Δ). By a simple reduction, the latter lower bounds also hold for the construction of maximal matchings and maximal independent sets.

Polynomial-time data reduction for dominating set

Abstract: Dealing with the NP-complete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy-to-implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.

A New Heuristic for the Minimum Connected Dominating Set Problem on Ad Hoc Wireless Networks

Abstract: Given a graph G = (V, E), a dominating set D is a subset of V such that any vertex not in D is adjacent to at least one vertex in D. Efficient algorithms for computing the minimum connected dominating set (MCDS) are essential for solving many practical problems, such as finding a minimum size backbone in ad hoc networks. Wireless ad hoc networks appear in a wide variety of applications, including mobile commerce, search and discovery, and military battlefield. In this chapter we propose a new efficient heuristic algorithm for the minimum connected dominating set problem. The algorithm starts with a feasible solution containing all vertices of the graph. Then it reduces the size of the CDS by excluding some vertices using a greedy criterion. We also discuss a distributed version of this algorithm. The results of numerical testing show that, despite its simplicity, the proposed algorithm is competitive with other existing approaches.

Exact (exponential) algorithms for the dominating set problem

Abstract: We design fast exact algorithms for the problem of computing a minimum dominating set in undirected graphs. Since this problem is NP-hard, it comes with no big surprise that all our time complexities are exponential in the number n of vertices. The contribution of this paper are ‘nice’ exponential time complexities that are bounded by functions of the form cn with reasonably small constants c<2: For arbitrary graphs we get a time complexity of 1.93782n. And for the special cases of split graphs, bipartite graphs, and graphs of maximum degree three, we reach time complexities of 1.41422n, 1.73206n, and 1.51433n, respectively.

Structural and algorithmic aspects of massive social networks

Abstract: We study the algorithmic and structural properties of very large, realistic social contact networks. We consider the social network for the city of Portland, Oregon, USA, developed as a part of the TRANSIMS/EpiSims project at the Los Alamos National Laboratory. The most expressive social contact network is a bipartite graph, with two types of nodes: people and locations; edges represent people visiting locations on a typical day. Three types of results are presented. (i) Our empirical results show that many basic characteristics of the dataset are well-modeled by a random graph approach suggested by Fan Chung Graham and Lincoln Lu (the CL-model), with a power-law degree distribution. (ii) We obtain fast approximation algorithms for computing basic structural properties such as clustering coefficients and shortest paths distribution. We also study the dominating set problem for such networks; this problem arose in connection with optimal sensor-placement for disease-detection. We present a fast approximation algorithm for computing near-optimal dominating sets. (iii) Given the close approximations provided by the CL-model to our original dataset and the large data-volume, we investigate fast methods for generating such random graphs. We present methods that can generate such a random network in near-linear time, and show that these variants asymptotically share many key features of the CL-model, and also match the Portland social network.The structural results have been used to study the impact of policy decisions for controlling large-scale epidemics in urban environments.

A structural view on parameterizing problems: Distance from triviality

Abstract: Based on a series of known and new examples, we propose the generalized setting of “distance from triviality” measurement as a reasonable and prospective way of determining useful structural problem parameters in analyzing computationally hard problems. The underlying idea is to consider tractable special cases of generally hard problems and to introduce parameters that measure the distance from these special cases. In this paper we present several case studies of distance from triviality parameterizations (concerning Clique, Power Dominating Set, Set Cover, and Longest Common Subsequence) that exhibit the versatility of this approach to develop important new views for computational complexity analysis.

Some remarks on domination

Abstract: We prove a conjecture of Favaron et al. that every graph of order n and minimum degree at least three has a total dominating set of size at least n-2. We also present several related results about: (1) extentions to graphs of minimum degree two, (2) examining graphs where the bound is tight, and (3) a type of bipartite domination and its relation to transversals in hypergraphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 207–210, 2004

Fast Parameterized Algorithms for Graphs on Surfaces: Linear Kernel and Exponential Speed-Up

Abstract: Preprocessing by data reduction is a simple but powerful technique used for practically solving different network problems. A number of empirical studies shows that a set of reduction rules for solving Dominating Set problems introduced by Alber, Fellows & Niedermeier leads efficiently to optimal solutions for many realistic networks. Despite of the encouraging experiments, the only class of graphs with proven performance guarantee of reductions rules was the class of planar graphs. However it was conjectured in that similar reduction rules can be proved to be efficient for more general graph classes like graphs of bounded genus. In this paper we (i) prove that the same rules, applied to any graph G of genus g, reduce the k-dominating set problem to a kernel of size O(k+g), i.e. linear kernel. This resolves a basic open question on the potential of kernel reduction for graph domination. (ii) Using such a kernel we improve the best so far algorithm for k-dominating set on graphs of genus ≤ g from \(2^{O(g\sqrt{k}+g^{2})}n^{O(1)}\) to \(2^{O(\sqrt{gk}+g)}+n^{O(1)}\). (iii) Applying tools from the topological graph theory, we improve drastically the best so far combinatorial bound to the branchwidth of a graph in terms of its minimum dominating set and its genus. Our new bound provides further exponential speed-up of our algorithm for the k-dominating set and we prove that the same speed-up applies for a wide category of parameterized graph problems such as k-vertex cover, k-edge dominating set, k-vertex feedback set, k-clique transversal number and several variants of the k-dominating set problem. A consequence of our results is that the non-parameterized versions of all these problems can be solved in subexponential time when their inputs have sublinear genus.

Tight lower bounds for certain parameterized NP-hard problems

Abstract: Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time n/sup o(k)/poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t - l)-st level W[t $1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted SAT, dominating set, hitting set, set cover, and feature set, cannot be solved in time n/sup o(k)/poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W[l] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-SAT (for any fixed q /spl ges/ 2), clique, and independent set, cannot be solved in time n/sup o(k)/ unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n/sup k/ poly(m) or O(n/sup k/).

Distributed algorithms for coloring and domination in wireless ad hoc networks

Abstract: We present fast distributed algorithms for coloring and (connected) dominating set construction in wireless ad hoc networks. We present our algorithms in the context of Unit Disk Graphs which are known to realistically model wireless networks. Our distributed algorithms take into account the loss of messages due to contention from simultaneous interfering transmissions in the wireless medium. We present randomized distributed algorithms for (conflict-free) Distance-2 coloring, dominating set construction, and connected dominating set construction in Unit Disk Graphs. The coloring algorithm has a time complexity of O(Δ log2n) and is guaranteed to use at most O(1) times the number of colors required by the optimal algorithm. We present two distributed algorithms for constructing the (connected) dominating set; the former runs in time O(Δ log 2n) and the latter runs in time O(log 2n). The two algorithms differ in the amount of local topology information available to the network nodes. Our algorithms are geared at constructing Well Connected Dominating Sets (WCDS) which have certain powerful and useful structural properties such as low size, low stretch and low degree. In this work, we also explore the rich connections between WCDS and routing in ad hoc networks. Specifically, we combine the properties of WCDS with other ideas to obtain the following interesting applications: An online distributed algorithm for collision-free, low latency, low redundancy and high throughput broadcasting. Distributed capacity preserving backbones for unicast routing and scheduling.

Distributed algorithms for coloring and domination in wireless ad hoc networks

Abstract: We present fast distributed algorithms for coloring and (connected) dominating set construction in wireless ad hoc networks. We present our algorithms in the context of Unit Disk Graphs which are known to realistically model wireless networks. Our distributed algorithms take into account the loss of messages due to contention from simultaneous interfering transmissions in the wireless medium. We present randomized distributed algorithms for (conflict-free) Distance-2 coloring, dominating set construction, and connected dominating set construction in Unit Disk Graphs. The coloring algorithm has a time complexity of O(Δ log 2 n) and is guaranteed to use at most O(1) times the number of colors required by the optimal algorithm. We present two distributed algorithms for constructing the (connected) dominating set; the former runs in time O(Δ log 2 n) and the latter runs in time O(log 2 n). The two algorithms differ in the amount of local topology information available to the network nodes. Our algorithms are geared at constructing Well Connected Dominating Sets (WCDS) which have certain powerful and useful structural properties such as low size, low stretch and low degree. In this work, we also explore the rich connections between WCDS and routing in ad hoc networks. Specifically, we combine the properties of WCDS with other ideas to obtain the following interesting applications: - An online distributed algorithm for collision-free, low latency, low redundancy and high throughput broadcasting. - Distributed capacity preserving backbones for unicast routing and scheduling.

Approximation Hardness of Dominating Set Problems

Abstract: We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. We state the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. For most of dominating set problems we prove asymptotically almost tight lower bounds. The results are applied to improve the lower bounds for other related problems such as the Maximum Induced Matching problem and the Maximum Leaf Spanning Tree problem.

Power and cost aware localized routing with guaranteed delivery in unit graph based ad hoc networks

Abstract: In a localized routing algorithm, each node currently holding a message makes forwarding decision solely based on the position information about itself, its neighbors and destination. In a unit graph, two nodes can communicate if and only if the distance between them is no more than the transmission radius, which is the same for each node. This paper proposes localized routing algorithms, aimed at minimizing total power for routing a message or maximizing the total number of routing tasks that a network can perform before a partition. The algorithms are combinations of known greedy power and/or cost aware localized routing algorithms and an algorithm that guarantees delivery. A shortcut procedure is introduced in later algorithm to enhance its performance. Another improvement is to restrict the routing to nodes in a dominating set. These improvements require two-hop knowledge at each node. The efficiency of proposed algorithms is verified experimentally by comparing their power savings, and the number of routing tasks a network can perform before a node loses all its energy, with the corresponding shortest weighted path algorithms and localized algorithms that use fixed transmission power at each node. Significant energy savings are obtained, and feasibility of applying power and cost-aware localized schemes is demonstrated. Copyright © 2004 John Wiley & Sons, Ltd.

Comments and corrections to "Dominating sets and neighbor elimination-based broadcasting algorithms in wireless networks"

Abstract: The paper by I. Stojmenovic et al. (2002) generated a lot of interest among researchers in ad hoc networks. A number of researchers questioned, through their articles, or directly to the first author, the correctness of the described procedure, and the correctness of the claim that the procedure does not need any communication exchange between nodes, in addition to "hello" messages needed to learn information about neighboring nodes. This correspondence completes the article by providing the actual dominating set definitions used in the procedure (from which zero communication overhead follows easily), the correct procedure (the published one has few misprints at key places), and the proof that the new definitions and procedure indeed define connected dominating sets.

Fast parallel molecular solution to the dominating-set problem on massively parallel bio-computing

Abstract: This paper shows how to use DNA strands to construct solution space of molecules for the dominating-set problem and how to apply biological operations to solve the problem from the solution space of molecules. In order to achieve this, we have proposed some DNA based parallel algorithms using the operations in Adleman-Lipton model, together with the analysis of the computational complexity for DNA parallel algorithms.

Weakly convex and convex domination numbers

Abstract: Two new domination parameters for a connected graph \(G\): the weakly convex domination number of \(G\) and the convex domination number of \(G\) are introduced. Relations between these parameters and the other domination parameters are derived. In particular, we study for which cubic graphs the convex domination number equals the connected domination number.

The bidimensional theory of bounded-genus graphs

Abstract: Bidimensionality is a powerful tool for developing subexponential fixed-parameter algorithms for combinatorial optimization problems on graph families that exclude a minor. This paper completes the theory of bidimensionality for graphs of bounded genus (which is a minor-excluding family). Specifically we show that, for any problem whose solution value does not increase under contractions and whose solution value is large on a grid graph augmented by a bounded number of handles, the treewidth of any bounded-genus graph is at most a constant factor larger than the square root of the problem’s solution value on that graph. Such bidimensional problems include vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, r-dominating set, connected dominating set, planar set cover, and diameter. This result has many algorithmic and combinatorial consequences. On the algorithmic side, by showing that an augmented grid is the prototype bounded-genus graph, we generalize and simplify many existing algorithms for such problems in graph classes excluding a minor. On the combinatorial side, our result is a step toward a theory of graph contractions analogous to the seminal theory of graph minors by Robertson and Seymour.

The Domatic Number of Regular Graphs.

Abstract: The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and maximum degree. As a corollary, we obtain the result that the domatic number of an r-regular graph is at least (r + 1)/(3ln(r + 1)).

A characterization of trees with equal total domination and paired-domination numbers.

Abstract: Let G = (V,E) be a graph without isolated vertices. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to at least one vertex in S. A total dominating set S ⊆ V is a paired-dominating set if the induced subgraph G[S] has at least one perfect matching. The paired-domination number γpr(G) is the minimum cardinality of a paired-domination set of G. In this paper, we provide a constructive characterization of those trees with equal total domination and paired-domination numbers, and of those trees for which the paired domination number is twice the matching number. ∗ Research of the first and second authors supported by the National Nature Science Foundation of China under grant 10101010 and the Young Science Foundation of Shanghai Education Committee under grant 01QN6262. † Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal. 32 ERFANG SHAN, LIYING KANG AND MICHAEL HENNING

Radio Network Clustering from Scratch

Abstract: We propose a novel randomized algorithm for computing a dominating set based clustering in wireless ad-hoc and sensor networks. The algorithm works under a model which captures the characteristics of the set-up phase of such multi-hop radio networks: asynchronous wake-up, the hidden terminal problem, and scarce knowledge about the topology of the network graph. When modelling the network as a unit disk graph, the algorithm computes a dominating set in polylogarithmic time and achieves a constant approximation ratio.

Reducing connected dominating set size with multipoint relays in ad hoc wireless networks

Abstract: Broadcasting is an important communication mechanism in ad hoc wireless networks. The simplest way to do broadcasting is pure flooding, in which each node retransmits a packet after receiving it, thus generates many redundant retransmissions. The rule based on dominating sets can reduce the number of retransmissions. A dominating set is a set of nodes such that any node in the network is a neighbor of some element in the set. However, computing a minimum size connected dominating set is NP hard. Several existing algorithms use the idea of multipoint relays to reduce the size of the connected dominating set. The authors of this paper observed that these algorithms can be further improved. Thus, three improvements are introduced here. It is proved that these improvements can compute a connected dominating set of the network, and simulation results show that they can further reduce the size of the dominating set. Also, extensions to power-aware broadcasting algorithms are discussed.

Efficient computation of maximal independent sets in unstructured multi-hop radio networks

Abstract: When being deployed, ad-hoc and sensor networks are unstructured and lack an efficient and reliable communication scheme. Hence, the organization of a MAC layer is the primary goal during and immediately after the deployment of such networks. Computing a good initial clustering facilitates this task and is therefore a vital part of the initialization process. A clustering based on a maximal independent set provides several highly desirable properties. Besides yielding a dominating set of good quality, such a clustering avoids interference between clusterheads, thus allowing efficient communication. We propose a novel algorithm that works under a model capturing the characteristics of the initialization phase of unstructured radio networks, i.e., asynchronous wake-up, scarce knowledge about the topology of the network graph, no collision detection, and the hidden terminal problem. We show that even under these hard conditions, the algorithm computes a maximal independent set in polylogarithmic time.