Conference Paper

# Approximation of Rectangle Stabbing and Interval Stabbing Problems.

DOI: 10.1007/978-3-540-30140-0_39 Conference: Algorithms - ESA 2004, 12th Annual European Symposium, Bergen, Norway, September 14-17, 2004, Proceedings
Source: DBLP

ABSTRACT

The weighted rectangle multi-stabbing problem (WRMS) can be described as follows: given is a grid in IR2\mathop{I\!\!R}^2consisting of columns and rows each having a positive integral weight, and a set of closed axis-parallel rectangles each having
a positive integral demand. The rectangles are placed arbitrarily in the grid with the only assumption that each rectangle
is intersected by at least one column and at least one row. The objective is to find a minimum weight (multi)set of columns
and rows of the grid so that for each rectangle the total multiplicity of selected columns and rows stabbing it is at least
its demand. (A column or row is said to stab a rectangle if it intersects it.)

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• "Kovaleva and Spieksma [16] recently studied several variants of rectangle stabbing. The group cut on a path problem was introduced by Hassin and Segev [12]. "
##### Chapter: Rounding to an Integral Program
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ABSTRACT: We present a general framework for approximating several NP-hard problems that have two underlying properties in common. First, the problems we consider can be formulated as integer covering programs, possibly with additional side constraints. Second, the number of covering options is restricted in some sense, although this property may be well hidden. Our method is a natural extension of the threshold rounding technique.
05/2005: pages 44-54;
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##### Article: Approximation Algorithms for Rectangle Piercing Problems
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ABSTRACT: Abstract Piercing problems arise often in facility location, which is a well-studied area of computational geometry. The general form of the piercing problem discussed in this dissertation asks for the minimum number of facilities for a set of given rectangular demand regions such that each region has at least one facility located within it. It has been shown that even if all regions are uniform sized squares, the problem is NP-hard. Therefore we concentrate on approximation algorithms for the problem. As the known approximation ratio for arbitrarily sized rectangles is poor, we restrict our efiort to designing approximation algorithms for unit-height rectangles. Our †-approximation scheme requires n,(1=†
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##### Article: Approximation Algorithms for Rectangle Stabbing and Interval Stabbing Problems
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ABSTRACT: In the weighted rectangle stabbing problem we are given a grid in ℝ 2 consisting of columns and rows each having a positive integral weight, and a set of closed axis-parallel rectangles each having a positive integral demand. The rectangles are placed arbitrarily in the grid with the only assumption being that each rectangle is intersected by at least one column or row. The objective is to find a minimum-weight (multi)set of columns and rows of the grid so that for each rectangle the total multiplicity of selected columns and rows stabbing it is at least its demand. A special case of this problem, called the interval stabbing problem, arises when each rectangle is intersected by exactly one row. We describe an algorithm called STAB, which is shown to be a constant-factor approximation algorithm for different variants of this stabbing problem.
SIAM Journal on Discrete Mathematics 01/2006; 20(3):748-768. DOI:10.1137/S089548010444273X · 0.65 Impact Factor