Given a range space (X,R), where R § 2 X, the hitting set problem is to find a smallest-cardinality subset H § X that intersects each set in R. We present near-linear-time approximation algorithms for the hitting set problem in the following geometric settings: (i) R is a set of planar regions with small union complexity. (ii) R is a set of axis-parallel d-dimensional boxes in Rd . In both cases X is either the entire R d , or a finite set of points in R d . The approximation factors yielded by the algorithm are small; they are either the same as, or within very small factors off the best factors known to be computable in polynomial time.
Bibliographical noteFunding Information:
Work on this paper has been supported by a joint grant, no. 2006/194, from the US-Israel Binational Science Foundation. Work by Pankaj Agarwal is also supported by NSF under grants CNS-05-40347, CCF-06 -35000, IIS-07-13498, and CCF-09-40671, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, by an NIH grant 1P50-GM-08183-01, and by a DOE grant OEG-P200A070505. Work by Micha Sharir has also been supported by NSF Grants CCF-05-14079 and CCF-08-30272, by Grants 155/05 and 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.
- Approximation algorithms
- Geometric range spaces
- Hitting sets