Domination in Geometric Intersection Graphs

For intersection graphs of disks and other fat objects, polynomial-time approximation schemes are known for the independent set and vertex cover problems, but the existing techniques were not able to deal with the dominating set problem except in the special case of unit-size objects. We present approximation algorithms and inapproximability results that shed new light on the approximability of the dominating set problem in geometric intersection graphs. On the one hand, we show that for intersection graphs of arbitrary fat objects, the dominating set problem is as hard to approximate as for general graphs. For intersection graphs of arbitrary rectangles, we prove APX-hardness. On the other hand, we present a new general technique for deriving approximation algorithms for various geometric intersection graphs, yielding constant-factor approximation algorithms for r-regular polygons, where r is an arbitrary constant, for pairwise homothetic triangles, and for rectangles with bounded aspect ratio. For arbitrary fat objects with bounded ply, we get a (3 + Ɛ)-approximation algorithm.

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