Geometric covering is a well-studied topic in computational geometry. We study three covering problems: Disjoint Unit-Disk Cover, Depth-(≤ K) Packing and Red-Blue Unit-Square Cover.
In the Disjoint Unit-Disk Cover problem, we are given a point set and want to cover the maximum number of points using disjoint unit disks. We prove that the problem is NP-complete and give a polynomial-time approximation scheme (PTAS) for it.
In Depth-(≤ K) Packing for Arbitrary-Size Disks/Squares, we are given a set of arbitrary-size disks/squares, and want to find a subset with depth at most K and maximizing the total area. We prove a depth reduction theorem and present a PTAS.
In Red-Blue Unit-Square Cover, we are given a red point set, a blue point set and a
set of unit squares, and want to find a subset of unit squares to cover all the blue points and the minimum number of red points. We prove that the problem is NP-hard, and give a PTAS for it. A "mod-one" trick we introduce can be applied to several other covering problems on unit squares.
| Identifer | oai:union.ndltd.org:WATERLOO/oai:uwspace.uwaterloo.ca:10012/7703 |
| Date | January 2013 |
| Creators | Hu, Nan |
| Source Sets | University of Waterloo Electronic Theses Repository |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
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