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New approximations for minimum-weighted dominating sets and minimum-weighted connected dominating sets on unit disk graphs. (English) Zbl 1209.68389
Summary: Given a node-weighted graph, the Minimum-Weighted Dominating Set (MWDS) problem is to find a minimum-weighted vertex subset such that, for any vertex, it is contained in this subset or it has a neighbor contained in this set. And the Minimum-Weighted Connected Dominating Set (MWCDS) problem is to find a MWDS such that the graph induced by this subset is connected. In this paper, we study these two problems on a unit disk graph. A $$(4 +\varepsilon )$$-approximation algorithm for an MWDS based on a dynamic programming algorithm for a Min-Weight Chromatic Disk Cover is presented. Meanwhile, we also propose a $$(1 +\varepsilon )$$-approximation algorithm for the connecting part by showing a polynomial-time approximation scheme for a Node-Weighted Steiner Tree problem when the given terminal set is $$c$$-local and thus obtain a $$(5+\varepsilon)$$-approximation algorithm for an MWCDS.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 68W25 Approximation algorithms
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