Complexity and Approximability of Hyperplane
Covering Problems
Michael Khachay
⋆
Krasovsky Institute of Mathematics and Mechanics, UB RAS,
16 S. Kovalevskoy st., Ekaterinburg, Russia
Ural Federal University, 19 Mira st., Ekaterinburg, Russia
Omsk State Technical University, 11 Mira ave, Omsk, Russia
Abstract. The well known N.Megiddo complexity result for Point Cover Prob-
lem on the plane is extended onto d-dimensional space (for any fixed d > 1).
It is proved that Min-dPC problem is L-reducible to Min-(d + 1)PC problem,
therefore for any fixed d > 1 there is no PTAS for Min-dPC problem, unless
P = NP.
1 Introduction
Settings of geometric covering problem and related problems are usual in various oper-
ations research domains [1-3]: optimal facility location theory, cluster analysis, pattern
recognition, etc. Mathematically, family of such problems can be partition into two
classes.
The first one contains special cases and modifications of well-known abstract Set
Cover problem. The main general feature shared by these problems is the finiteness of
the initial family of subsets, for which it is required to find a subfamily (or just prove its
existence) covering some target set and satisfying given optimality conditions. There
are many papers studying problems from this class (see survey at [4]). The classical
papers [5-7] seem to be the most important among them. First two papers contain in-
tractability proof of Set Cover problem and two main design patterns for constructing
approximationalgorithms for this problem. The last paper proves the optimality of these
patterns, unless P = NP.
The second class consists of problems without the mentioned above finiteness con-
straint. Usually, the initial family of subsets is given here implicitly in terms of some
geometric property characterizing its elements. For instance, for a givenset it is required
to find a minimal cardinality cover by straight lines, circles of a given radii, etc.
2 Point Cover (2PC) Problem
In the paper, a series of hyperplane covering problems for given finite sets in finite-
dimensional vector spaces of fixed dimension d > 1 is considered. The first element of
⋆
This research was partially supported by RFBR, grants no. 13-07-00181 and 13-01-210, and
Ural Branch of RAS, grants no. 12-P-1-1016 and 12-S-1017/1
Khachay M..
Complexity and Approximability of Hyperplane Covering Problems.
DOI: 10.5220/0004394601090113
In Proceedings of the 4th International Workshop on Image Mining. Theory and Applications (IMTA-4-2013), pages 109-113
ISBN: 978-989-8565-50-1
Copyright
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2013 SCITEPRESS (Science and Technology Publications, Lda.)