Showing papers on "Dominating set published in 2002"

Dominating sets and neighbor elimination-based broadcasting algorithms in wireless networks

Abstract: In a multihop wireless network, each node has a transmission radius and is able to send a message to all of its neighbors that are located within the radius. In a broadcasting task, a source node sends the same message to all the nodes in the network. In this paper, we propose to significantly reduce or eliminate the communication overhead of a broadcasting task by applying the concept of localized dominating sets. Their maintenance does not require any communication overhead in addition to maintaining positions of neighboring nodes. Retransmissions by only internal nodes in a dominating set is sufficient for reliable broadcasting. Existing dominating sets are improved by using node degrees instead of their ids as primary keys. We also propose to eliminate neighbors that already received the message and rebroadcast only if the list of neighbors that might need the message is nonempty. A retransmission after negative acknowledgements scheme is also described. The important features of the proposed algorithms are their reliability (reaching all nodes in the absence of message collisions), significant rebroadcast savings, and their localized and parameterless behavior. The reduction in communication overhead for the broadcasting task is measured experimentally. Dominating set based broadcasting, enhanced by a neighbor elimination scheme and highest degree key, provides reliable broadcast with /spl les/53 percent of node retransmissions (on random unit graphs with 100 nodes) for all average degrees d. Critical d is around 4, with <48 percent for /spl les/3, /spl les/40 percent for d/spl ges/10, and /spl les/20 percent for d/spl ges/25. The proposed methods are better than existing ones in all considered aspects: reliability, rebroadcast savings, and maintenance communication overhead. In particular, the cluster structure is inefficient for broadcasting because of considerable communication overhead for maintaining the structure and is also inferior in terms of rebroadcast savings.

Domination in Graphs Applied to Electric Power Networks

Abstract: The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graphs. We consider the graph theoretical representation of this problem as a variation of the dominating set problem and define a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph G is the power domination number $\gamma_P(G)$. We show that the power dominating set (PDS) problem is NP-complete even when restricted to bipartite graphs or chordal graphs. On the other hand, we give a linear algorithm to solve the PDS for trees. In addition, we investigate theoretical properties of $\gamma_P(T)$ in trees T.

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Abstract: . We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (G) n) time. The same technique can be used to show that the k -FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c 1 ^ \sqrt k n) time, where c 1 =3^ 36\sqrt 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k -DOMINATING SET, e.g., k -INDEPENDENT DOMINATING SET and k -WEIGHTED DOMINATING SET.

Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks

Abstract: We present a series of approximation algorithms for finding a small weakly-connected dominating set (WCDS) in a given graph to be used in clustering mobile ad hoc networks. The structure of a graph can be simplified using WCDS's and made more succinct for routing in ad hoc networks. The theoretical performance ratio of these algorithms is O(ln Δ) compared to the minimum size WCDS, where Δ is the maximum degree of the input graph. The first two algorithms are based on the centralized approximation algorithms of Guha and Khuller cite guha-khuller-1998 for finding small connected dominating sets (CDS's). The main contribution of this work is a completely distributed algorithm for finding small WCDS's and the performance of this algorithm is shown to be very close to that of the centralized approach. Comparisons between our work and some previous work (CDS-based) are also given in terms of the size of resultant dominating sets and graph connectivity degradation.

An efficient distributed algorithm for constructing small dominating sets

Abstract: The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in O(log n log Δ) rounds with high probability, where n is the number of nodes, Δ is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within O (log Δ) of the optimal in expectation and within O(log n) of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements.

Extended dominating-set-based routing in ad hoc wireless networks with unidirectional links

Abstract: We extend dominating-set-based routing to networks with unidirectional links. Specifically, an efficient localized algorithm for determining a dominating and absorbant set of vertices (mobile hosts) is given and this set can be easily updated when the network topology changes dynamically. A host /spl nu/ is called a dominating neighbor (absorbant neighbor) of another host u if there is a directed edge from /spl nu/ to u (from u to /spl nu/). A subset of vertices is dominating and absorbant if every vertex not in the subset has one dominating neighbor and one absorbant neighbor in the subset. The derived dominating and absorbant set exhibits good locality properties; that is, the change of a node status (dominating/dominated) affects only the status of nodes in the neighborhood. The notion of dominating and absorbant set can also be applied iteratively on the dominating and absorbant set itself, forming a hierarchy of dominating and absorbant sets. The effectiveness of our approach is confirmed and the locality of node status update is verified through simulation.

Internal Node and Shortcut Based Routing with Guaranteed Delivery in Wireless Networks

Abstract: Several localized position based routing algorithms for wireless networks were described recently. In greedy routing algorithm (that has close performance to the shortest path algorithm, if successful), sender or node i>S currently holding the message i>m forwards i>m to one of its neighbors that is the closest to destination. The algorithm fails if i>S does not have any neighbor that is closer to destination than i>S. i>FACE algorithm guarantees the delivery of i>m if the network, modeled by unit graph, is connected. i>GFG algorithm combines greedy and i>FACE algorithms. Greedy algorithm is applied as long as possible, until delivery or a failure. In case of failure, the algorithm switches to i>FACE algorithm until a node closer to destination than last failure node is found, at which point greedy algorithm is applied again. Past traffic does not need to be memorized at nodes. In this paper we further improve the performance of i>GFG algorithm, by reducing its average hop count. First we improve the i>FACE algorithm by adding a sooner-back procedure for earlier escape from i>FACE mode. Then we perform a i>shortcut procedure at each forwarding node i>S. Node i>S uses the local information available to calculate as many hops as possible and forwards the packet to the last known hop directly instead of forwarding it to the next hop. The second improvement is based on the concept of dominating sets. Each node in the network is classified as internal or not, based on geographic position of its neighboring nodes. The network of internal nodes defines a connected dominating set, i.e., and each node must be either internal or directly connected to an internal node. In addition, internal nodes are connected. We apply several existing definitions of internal nodes, namely the concepts of intermediate, inter-gateway and gateway nodes. We propose to run i>GFG routing, enhanced by shortcut procedure, on the dominating set, except possibly the first and last hops. The performance of proposed algorithms is measured by comparing its average hop count with hop count of the basic i>GFG algorithm and the benchmark shortest path algorithm, and very significant improvements were obtained for low degree graphs. More precisely, we obtained localized routing algorithm that guarantees delivery and has very low excess in terms of hop count compared to the shortest path algorithm. The experimental data show that the length of additional path (in excess of shortest path length) can be reduced to about half of that of existing i>GFG algorithm.

Improved Tree Decomposition Based Algorithms for Domination-like Problems

Abstract: We present an improved dynamic programming strategy for DOMINATING set and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4kn) algorithm for dominating set, where n is the number of nodes of the tree decomposition. This result improves the previously best known algorithm of Telle and Proskurowski running in time O(9kn). The key to our result is an argument on a certain "monotonicity" in the table updating process during dynamic programming.Moreover, various other domination-like problems as discussed by Telle and Proskurowski are treated with our technique.We gain improvements on the base of the exponential term in the running time ranging between 55% and 68% in most of these cases. These results mean significant breakthroughs concerning practical implementations.

A characterization of roman trees

Abstract: A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of f is w(f) = ∑ v∈V f(v). The Roman domination number is the minimum weight of an RDF in G. It is known that for every graph G, the Roman domination number of G is bounded above by twice its domination number. Graphs which have Roman domination number equal to twice their domination number are called Roman graphs. At the Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms, and Applications held at Western Michigan University in June 2000, Stephen T. Hedetniemi in his principal talk entitled “Defending the Roman Empire” posed the open problem of characterizing the Roman trees. In this paper, we give a characterization of Roman trees.

1-perfect codes in Sierpiński graphs

Abstract: Sierpinski graphs S (n, κ) generalise the Tower of Hanoi graphs—the graph S (n, 3) is isomorphic to the graph Hn of the Tower of Hanoi with n disks. A 1-perfect code (or an efficient dominating set) in a graph G is a vertex subset of G with the property that the closed neighbourhoods of its elements form a partition of V (G). It is proved that the graphs S (n, κ) possess unique 1-perfect codes, thus extending a previously known result for Hn. An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to Hn.

Dominating set based bluetooth scatternet formation with localized maintenance

Abstract: This paper addresses the problem of scatternet formation and maintenance for multi-hop Bluetooth based personal area and ad hoc networks with minimal communication overhead. Each node is assumed to know its position and position of all its neighbours. The proposed formation algorithms have three phases. In the first phase the unit graph is constructed (each node establishes connection with all its neighbors that are located within its transmission radius, which is equal for all nodes), and, if planar structure is desirable, localized sparse subgraph (such as relative neighbourhood or Gabriel graph) is extracted. In the second phase, the degree of each node is limited to 7 by applying Yao subgraph construct simultaneously on all nodes with excessive degree, followed by either elimination of directed edges or the application of reverse Yao construct. In the last phase, master-slave relations are created by applying higher degree priority (with dominating set membership as the primary key). The creation and maintenance requires minimal overhead in addition to maintaining accurate location information for one-hop neighbours. The proposed schemes have localized maintenance property (scatternet maintenance due to movement or activity change of a single node is limited to the locality of that node), which is not the case with the existing clustering based Bluetooth scatternet formation schemes.

Improved Parameterized Algorithms for Planar Dominating Set

Abstract: Recently, there has been a lot of interest and progress in lowering the worst-case time complexity for the PLANAR DOMINATING SET problem. In this paper, we present improved parameterized algorithms for the PLANAR DOMINATING SET problem. In particular, given a planar graph G and a positive integer k, we can compute a dominating set of size bounded by k or report that no such set exists in time O(227?kn), where n is the number of vertices in G. Our algorithms induce a significant improvement over the previous best algorithm for the problem.

Improved parameterized algorithms for planar dominating set

Abstract: Recently, there has been a lot of interest and progress in lowering the worst-case time complexity for the PLANAR DOMINATING SET problem. In this paper, we present improved parameterized algorithms for the PLANAR DOMINATING SET problem. In particular, given a planar graph G and a positive integer k, we can compute a dominating set of size bounded by k or report that no such set exists in time O(2 27 n), where n is the number of vertices in G. Our algorithms induce a significant improvement over the previous best algorithm for the problem.

Augmenting Trees to Meet Biconnectivity and Diameter Constraints

Abstract: Given a graph G=(V,E) and a positive integer D , we consider the problem of finding a minimum number of new edges E' such that the augmented graph G'=(V,E\cup E') is biconnected and has diameter no greater than D. In this note we show that this problem is NP-hard for all fixed D , by employing a reduction from the DOMINATING SET problem. We prove that the problem remains NP-hard even for forests and trees, but in this case we present approximation algorithms with worst-case bounds 3 (for even D ) and 6 (for odd D ). A closely related problem of finding a minimum number of edges such that the augmented graph has diameter no greater than D has been shown to be NP-hard by Schoone et al. [21] when D=3 , and by Li et al. [17] when D=2.

Experimental Analysis of Heuristic Algorithms for the Dominating Set Problem

Abstract: We say a vertex v in a graph G covers a vertex w if v=w or if v and w are adjacent. A subset of vertices of G is a dominating set if it collectively covers all vertices in the graph. The dominating set problem, which is NP-hard, consists of finding a smallest possible dominating set for a graph. The straightforward greedy strategy for finding a small dominating set in a graph consists of successively choosing vertices which cover the largest possible number of previously uncovered vertices. Several variations on this greedy heuristic are described and the results of extensive testing of these variations is presented. A more sophisticated procedure for choosing vertices, which takes into account the number of ways in which an uncovered vertex may be covered, appears to be the most successful of the algorithms which are analyzed. For our experimental testing, we used both random graphs and graphs constructed by test case generators which produce graphs with a given density and a specified size for the smallest dominating set. We found that these generators were able to produce challenging graphs for the algorithms, thus helping to discriminate among them, and allowing a greater variety of graphs to be used in the experiments.

Efficient Data Reduction for DOMINATING SET: A Linear Problem Kernel for the Planar Case

Abstract: Dealing with the NP-complete Dominating Set problem on undirected 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 on planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. This answers an open question from previous work on the parameterized complexity of Dominating Set on planar graphs.

The Dominating Set Problem Is Fixed Parameter Tractable for Graphs of Bounded Genus

Abstract: We describe an algorithm for the dominating set problem withtime complexity O((24g2 + 24g + 1)kn2) for graphs of bounded genus g, where k is the size of the set. It has previously been shown that this problem is fixed parameter tractable for planar graphs. Our method is a refinement of the earlier techniques.

Minimum independent dominating sets of random cubic graphs

Abstract: We present a heuristic for finding a small independent dominating set D of cubic graphs. We analyze the performance of this heuristic, which is a random greedy algorithm, on random cubic graphs using differential equations, and obtain an upper bound on the expected size of D. A corresponding lower bound is derived by means of a direct expectation argument. We prove that D asymptotically almost surely satisfies 0.2641n ≤ |D| ≤ 0.27942n.

Dominating sets for rectilinear center location problems with polyhedral barriers

Abstract: In planar location problems with barriers one considers regions which are forbidden for the siting of new facilities as well as for trespassing. These problems are important since they model various actual applications. The resulting mathematical models have a nonconvex objective function and are therefore difficult to tackle using standard methods of location theory even in the case of simple barrier shapes and distance functions. For the case of center objectives with barrier distances obtained from the rectilinear or Manhattan metric, it is shown that the problem can be solved in polynomial time by identifying a dominating set. The resulting genuinely polynomial algorithm can be combined with bound computations which are derived from solving closely connected restricted location and network location problems. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 647–665, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10038

Server Placements, Roman Domination and other Dominating Set Variants

Abstract: Dominating sets in their many variations model a wealth of optimization problems like facility location or distributed file sharing. For instance, when a request can occur at any node in a graph and requires a server at that node, a minimum dominating set represents a minimum set of servers that serve an arbitrary single request by moving a server along at most one edge. This paper studies domination problems for two requests. For the problem of placing a minimum number of servers such that two requests at different nodes can be served with two different servers (called win-win), we present a logarithmic approximation, and we prove that nothing better is possible. We show that the same is true for Roman domination, the well studied problem variant that asks for each vertex to either possess its own server or to have a neighbor with two servers. Still the same is true if each idle server can move along one edge while the first of both requests is being served. For planar graphs, we propose a PTAS for Roman domination (and show that nothing better exists), and we get a constant approximation for win-win.

Fibonacci Polynomials and Parity Domination in Grid Graphs

Abstract: A non-empty set of vertices is called an even dominating set if each vertex in the graph is adjacent to an even number of vertices in the set (adjacency is reflexive). In this paper, the Fibonacci polynomials are studied over GF(2) with particular emphasis on their divisibility properties and their relation to the existence of even dominating sets in grid graphs and properties of a corresponding recurrence.

Scheduling for maximum capacity in SDMA/TDMA networks

Abstract: This paper presents a general information-theoretic framework for the problem of spatial and temporal packet scheduling in SDMA/TDMA systems. The said problem is described as a partition of a given set of users and SDMA and TDMA are applied within each subset and among difference subsets, respectively. The capacity of SDMA/TDMA systems is derived as a function of the partition scheme. Optimization of the packet scheduling can then be performed by maximizing the capacity. The paper also presents a graph-theoretical model for the optimization problem. It shows that that the optimization is to find a proper independent dominating set in an associated graph.

Dominating Pair Graphs

Abstract: A pair of vertices of a graph is called a dominating pair if the vertex set of every path between these two vertices is a dominating set of the graph. A graph is a weak dominating pair graph if it has a dominating pair. Further, a graph is called a dominating pair graph if each of its connected induced subgraphs is a weak dominating pair graph. Dominating pair graphs form a class of graphs containing interval, permutation, cocomparability, and asteroidal triple-free graphs. Our purpose is to study the structural properties of dominating pair graphs. Our main results are a polar theorem for the dominating pairs in weak dominating pair graphs and an existence theorem for minimum cardinality connected dominating sets that induce a simple path in connected dominating pair graphs of diameter not equal to three. Furthermore, we present a forbidden induced subgraph characterization of chordal dominating pair graphs.