Dominating set

About: Dominating set is a research topic. Over the lifetime, 4058 publications have been published within this topic receiving 72432 citations. The topic is also known as: dominating set problem.

On the hardness of approximating minimization problems

Abstract: We prove results indicating that it is hard to compute efficiently good approximate solutions to the Graph Coloring, Set Covering and other related minimization problems. Specifically, there is an e > 0 such that Graph Coloring cannot be approximated with ratio n e unless P = NP. Set Covering cannot be approximated with ratio c log n for any c < 1/4 unless NP is contained in DTIME(n poly log n ). Similar results follow for related problems such as Clique Cover, Fractional Chromatic Number, Dominating Set, and others

A Best Possible Heuristic for the k-Center Problem

Abstract: In this paper we present a 2-approximation algorithm for the k-center problem with triangle inequality. This result is “best possible” since for any δ < 2 the existence of δ-approximation algorithm would imply that P = NP. It should be noted that no δ-approximation algorithm, for any constant δ, has been reported to date. Linear programming duality theory provides interesting insight to the problem and enables us to derive, in O|E| log |E| time, a solution with value no more than twice the k-center optimal value. A by-product of the analysis is an O|E| algorithm that identifies a dominating set in G2, the square of a graph G, the size of which is no larger than the size of the minimum dominating set in the graph G. The key combinatorial object used is called a strong stable set, and we prove the NP-completeness of the corresponding decision problem.

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.

Approximation Algorithms for Connected Dominating Sets

Abstract: The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related 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 as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Δ)+2 and H(Δ)+2 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 at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn+1) \ln n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103 ). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ).