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Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. (English) Zbl 1148.05308
Díaz, Josep (ed.) et al., Approximation, randomization and combinatorial optimization. Algorithms and techniques. 9th international workshop on approximation algorithms for combinatorial optimization problems, APPROX 2006, and 10th international workshop on randomization and computation, RANDOM 2006, Barcelona, Spain, August 28–30, 2006. Proceedings. Berlin: Springer (ISBN 3-540-38044-2/pbk). Lecture Notes in Computer Science 4110, 3-14 (2006).
Summary: For a given graph with weighted vertices, the goal of the minimum-weight dominating set problem is to compute a vertex subset of smallest weight such that each vertex of the graph is contained in the subset or has a neighbor in the subset. A unit disk graph is a graph in which each vertex corresponds to a unit disk in the plane and two vertices are adjacent if and only if their disks have a non-empty intersection. We present the first constant-factor approximation algorithm for the minimum-weight dominating set problem in unit disk graphs, a problem motivated by applications in wireless ad-hoc networks. The algorithm is obtained in two steps: First, the problem is reduced to the problem of covering a set of points located in a small square using a minimum-weight set of unit disks. Then, a constant-factor approximation algorithm for the latter problem is obtained using enumeration and dynamic programming techniques exploiting the geometry of unit disks. Furthermore, we also show how to obtain a constant-factor approximation algorithm for the minimum-weight connected dominating set problem in unit disk graphs.
For the entire collection see [Zbl 1114.68006].

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C85 Graph algorithms (graph-theoretic aspects) 68W25 Approximation algorithms
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