A Robust PTAS for Maximum Weight Independent Sets in Unit Disk Graphs

Abstract. A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1 + ε)-approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects.

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We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input. The runtime of the PTAS is n O(1/εlog 1/ε). The algorithm accepts any undirected graph as input, and returns a (1 + ε)-approximate minimum dominating set, or a certificate showing that the input graph is no unit disk graph, making the algorithm robust. The PTAS can easily be adapted to other classes of geometric intersection graphs