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Algorithms on Graphs with Small Dominating Targets
Abstract
A dominating target of a graph G=(V,E) is a set of vertices T s.t. for all W ⊆ V, if T ⊆ W and induced subgraph on W is connected, then W is a dominating set of G. The size of the smallest dominating target is called dominating target number of the graph, dt(G). We provide polynomial time algorithms for minimum connected dominating set, Steiner set, and Steiner connected dominating set in dominating-pair graphs (i.e., dt(G)=2). We also give approximation algorithm for minimum connected dominating set with performance ratio 2 on graphs with small dominating targets. This is a significant improvement on appx ≤d(opt + 2) given by Fomin et.al. [2004] on graphs with small d-octopus.
Classification: Dominating target, d-octopus, Dominating set, Dominating-pair graph, Steiner tree.
... In a wired network, the multicast packets are forwarded along the tree edges, and so the multicast routing problem can be defined as a Steiner tree problem where the multicast group members are the terminals (leaf nodes) in the Steiner tree [2]. The problem of finding a Steiner tree is known to be NP-complete [3], even if links have unit cost [4]. It should be noted that in some multicast routing protocols [5,6] the minimum spanning tree problem, which is a well known approach for broadcasting, is also used to model the multicast routing problem. ...
... The total size of the SCDS constructed by Muhammad's algorithm is smaller than 2 |MIS| OPT 1 1, and the message and time complexity of this algorithm are 2| | and | | 1 · | | · log| | , respectively. Aggarwal et al. [4] proposed an algorithm for approximating the MSCDS in a dominating pair graph. The time complexity of the proposed algorithm is | | · | | , where | | and | |denotes the cardinality of the vertex set and terminal-set, respectively. ...
Multicast routing is an effective way to establish the group communication when the same message or the same stream of data needs to be sent to multiple receivers. Multicast routing has attracted a lot of attention in group oriented computing due to supporting data transmission from a single source node to multiple destinations concurrently. The advantage of multicast routing lies in its capability of reducing the communication cost and saving the network resources by sending only one copy of the message over the shared link leading to different destinations. Generally speaking, this survey classifies the multicast routing protocols into four categories based on the underlying routing structure: tree‐based, mesh‐based, hybrid and stateless multicast routing protocols. This survey summarizes the well‐known most recent protocols of each category and compares their objectives, performances, advantages and disadvantages. This survey study helps us to choose the best multicasting protocol for each network application with respect to the requirements of application. Keyword Multicast routing, mobile ad hoc network, tree‐based protocol, mesh‐ based protocol, on‐demand protocol, proactive protocol 1.
... The total size of the SCDS constructed by Muhammad's algorithm is smaller than 2(|M I S|+OPT−1)+1, and the message and time complexity of this algorithm are O(2 |V |) and O ((|V | − 1) · |E| · log |V |), respectively. Aggarwal et al. [9] proposed an algorithm for approximating the MSCDS in a dominating pair graph. The time complexity of the proposed algorithm is O(|V | 8 · |R|), where |V |and |R|denotes the cardinality of the vertex set and terminal-set, respectively. ...
... Indeed, in SCDS problem, a specified subset, R, of the vertices has to be dominated by the a connected dominating set. Finding the SCDS of the network graph is a well-known approach proposed for solving the multicast routing problem in wireless ad-hoc networks [2,3,6789 , where subset R comprises the multicast source and the multicast receivers. In this method, the SCDS includes the intermediate nodes by which the massage sent out by the multicast source is relayed. ...
In this paper, we first propose three centralized learning automata-based heuristic algorithms for approximating a near optimal
solution to the minimum weight Steiner connected dominating set (WSCDS) problem. Finding the Steiner connected dominating
set of the network graph is a promising approach for multicast routing in wireless ad-hoc networks. Therefore, we present
a distributed implementation of the last approximation algorithm proposed in this paper (Algorithm III) for multicast routing
in wireless mobile ad-hoc networks. The proposed WSCDS algorithms are compared with the well-known existing algorithms and
the obtained results show that Algorithm III outperforms the others both in terms of the dominating set size and running time.
Our simulation experiments also show the superiority of the proposed multicast routing algorithm over the best previous methods
in terms of the packet delivery ratio, multicast route lifetime, and end-to-end delay.
KeywordsSteiner connected dominating set–Multicast routing–Learning automata–Wireless mobile Ad-hoc networks
... The most efficient method is to forwarding the multicast packets along the tree edges, and so the multicast routing problem can be defined as a Steiner tree problem where the multicast group members are terminals (leaf nodes) in Steiner tree [1]. Finding the Steiner tree is known to be an NP‐hard in graph theory [2] even if links have unit cost [3]. However, the minimal spanning tree problem which is a well known approach for broadcasting has been also used in [4, 5] to model the multicast routing problem. ...
Recently, several studies have been conducted to design mobility-based multicast routing protocols for wireless mobile ad hoc networks (MANET). These protocols assume that the mobility parameters of the network are fixed, and so they cannot perform well under real MANET scenarios in which the mobility parameters of the hosts vary over time at random. Finding the optimal solution to the multicast routing problem is incredibly hard, if the mobility parameters are assumed to be random variables. This becomes more difficult when the probability distribution function of these random variables is assumed to be unknown. In this paper, we propose a weighted multicast routing algorithm for MANET in which the mobility parameters are supposed to be random variables with unknown distribution. In this method, the multicast routing problem is first transformed into an equivalent stochastic Steiner tree problem in which the random weight associated with a communication link is its expected duration time. Then, a learning automata-based algorithm is proposed for solving the proxy Steiner tree problem. The aim of the proposed algorithm is to find the most stable multicast route (with the maximum duration) against the host mobility. Experimental results confirm the superiority of the proposed method over the best existing mobility-based multicast routing protocols in terms of the packet delivery ratio, multicast route lifetime, control message overhead, and end-to-end delay.
An independent set of three vertices is called an asteroidal triple (AT) if there exists a path between any two of them that avoids the neighborhood of the third. Asteroidal triple-free (AT-free) graphs are very well-studied, but some of their various superclasses are not. We study two of these superclasses: hereditary dominating pair (HDP) graphs and diametral path graphs. We correct a mistake that has appeared in the literature claiming that the class of diametral path graphs are a superclass of HDP. More specifically, we show that a graph with a dominating shortest path does not necessarily contain a dominating diametral path. We say a graph is a strict dominating pair graph if it contains a dominating pair but has no dominating diametral path, and we show structural and algorithmic properties of these graphs. To study properties of HDP graphs, we introduce the notion of spread in asteroidal triples. Given a dominating pair, we show that all paths between this pair meet the common neighborhood of some pair from each asteroidal triple. We use these results to improve the best known time complexity for the recognition of chordal HDP graphs.
The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to at least one node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected. This problem arises in network testing, as well as in wireless communication.
Two polynomial time algorithms that achieve approximation factors of O(H(Δ)) are presented, where Δ is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the more general problem of finding a connected dominating set of a specified set of vertices and provide a 3 ln n approximation factor. To prove the bound we also develop an optimal approximation algorithm for the unit node weighted Steiner tree problem.
This paper uses periodic matrix multiplication to improve the time complexities for a number of graph problems. The time for finding a clique cutset in a graph is reduced from O(nm) to O(n2.69), the time for finding an asteroidal triple is reduced to O(n2.82), and the time for finding a star cutset, a two-pair, and a dominating pair is reduced from O(nm) to O(n2.79).It is also shown that each of these problems is at least as hard as one of three basic graph problems for which the best known algorithms run in time O(nm) and O(nα).
List of Figures. List of Tables. 1: Preliminaries. 1.1. Preface. 1.2. The Domain of Discourse: Routing in VLSI Physical Design. 1.3. Overview of the Book. 1.4. Acknowledgements. 2: Area. 2.1. Introduction. 2.2. Performance Bounds for MST-Based Strategies. 2.3. Iterated 1-Steiner (I1S). 2.4. Enhancing I1S Performance. 2.5. Practical Implementation Options for I1S. 2.6. On the Maximum MST Degree. 2.7. Steiner Trees in Graphs. 3: Delay. 3.1. Preliminaries. 3.2. Geometric Approaches to Delay Minimization. 3.3. Minimization of Actual Delay. 3.4. New Directions. 4: Skew. 4.1. Preliminaries. 4.2. An Early Matching-Based Approach. 4.3. DME: Exact Zero Skew with Minimum Wirelength. 4.4. Planar-Embeddable Trees. 4.5. Remarks. 5: Multiple Objectives. 5.1. Minimum Density Trees. 5.2. Multi-Weighted Graphs. 5.3. Prescribed-Width Routing. A: Appendix: Signal Delay Estimators. A.1. Basics. A.2. Accuracy and Fidelity. References. Author Index. Term Index.
An abstract is not available.
Preface 1. Basic Concepts 2. Perfection, Generalized Perfection, and Related Concepts 3. Cycles, Chords and Bridges 4. Models and Interactions 5. Vertex and Edge Orderings 6. Posets 7. Forbidden Subgraphs 8. Hypergraphs and Graphs 9. Matrices and Polyhedra 10. Distance Properties 11. Algebraic Compositions and Recursive Definitions 12. Decompositions and Cutsets 13. Threshold Graphs and Related Concepts 14. The Strong Perfect Graph Conjecture Appendix A. Recognition Appendix B. Containment Relationships Bibliography Index.
Many families of perfect graphs such as interval graphs, permutation graphs, trapezoid graphs and cocomparability graphs demonstrate a type of linear ordering of their vertex sets. These graphs are all subfamilies of a class of graphs called the asteroidal triple-free graphs. (An independent triple {x, y, z} is called an asteroidal triple (AT, for short) if between any pair in the triple there exists a path that avoids the neighbourhood of the third vertex.) In this paper we argue that the property of being AT-free is what is enforcing the linear ordering of the vertex sets. To justify this claim, we present various structural properties and characterizations of AT-free graphs.