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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    ABSTRACT: In the rectangle stabbing problem, we are given a set of axis parallel rectangles and a set of horizontal and vertical lines, and our goal is to find a minimum size subset of lines that intersect all the rectangles. In this article, we study the capacitated version of this problem in which the input includes an integral capacity for each line. The capacity of a line bounds the number of rectangles that the line can cover. We consider two versions of this problem. In the first, one is allowed to use only a single copy of each line (hard capacities), and in the second, one is allowed to use multiple copies of every line, but the multiplicities are counted in the size (or weight) of the solution (soft capacities). We present an exact polynomial-time algorithm for the weighted one dimensional case with hard capacities that can be extended to the one dimensional weighted case with soft capacities. This algorithm is also extended to solve a certain capacitated multi-item lot-sizing inventory problem with joint set-up costs. For the case of d-dimensional rectangle stabbing with soft capacities, we present a 3d-approximation algorithm for the unweighted case. For d-dimensional rectangle stabbing problem with hard capacities, we present a bi-criteria algorithm that computes 4d-approximate solutions that use at most two copies of every line. Finally, we present hardness results for rectangle stabbing when the dimension is part of the input and for a two-dimensional weighted version with hard capacities.
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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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