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.
|Title of host publication||Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX)|
|Number of pages||12|
|Publication status||Published - 2006|
Ambühl, C., Erlebach, T., Mihalák, M., & Nunkesser, M. (2006). Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Disk Graphs. In Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) (pp. 3-14) https://doi.org/10.1007/11830924_3