15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 3 - Diskrete Mathematik, Algorithmen
Dienstag, 18. September 2001, 18.00, Hörsaal 23


On Covering $ \mathbb{Z}$-Grid Points by Rectangles

Stefan Porschen, Universität Köln


A problem of combinatorial geometry is discussed: Cover a finite set of points lying on an integer grid in the Euclidean plane by regular rectangles such that the total area, circumference and number of rectangles used is minimized. This problem seems to be NP-hard, which is surely the case for related problems concerning covering points arbitrarily distributed in the plane. Treating the case of minimal rectangle side lengths $ k<\lambda$ (grid constant), we propose an exact deterministic algorithm based on set theoretic dynamic programming, which then is improved by exploiting the rectangular and underlying grid structure. We also discuss a variant given by a further parameter $ p$ bounding the maximal possible covering cardinality. For this, we are able to find a time bound by a polynomial of degree $ O(p)$. Generalizations to arbitrary values of $ k$ and arbitrary (finite) space dimensions are possible. (A version of this talk has been presented at the Cologne-Twente-Workshop 2001, an extended abstract of which may be found as Electronic Notes on Discrete Mathematics (ENDM, Elsevier, Vol.8, 2001).)

E-Mail: porschen@informatik.uni-koeln.de

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