Optimal Wiring on Rectangular Structure

2019

In the Anchored Rectangle Packing (ARP) problem, we are given a set of points P in the unit square [0,1]^2 and seek a maximum-area set of axis-aligned interior-disjoint rectangles S, each of which is anchored at a point p in P. In the most prominent variant - Lower-Left-Anchored Rectangle Packing (LLARP) - rectangles are anchored in their lower-left corner. Freedman [W. T. Tutte (Ed.), 1969] conjectured in 1969 that, if (0,0) in P, then there is a LLARP that covers an area of at least 0.5. Somewhat surprisingly, this conjecture remains open to this day, with the best known result covering an area of 0.091 [Dumitrescu and Toth, 2015]. Maybe even more surprisingly, it is not known whether LLARP - or any ARP-problem with only one anchor - is NP-hard. In this work, we first study the Center-Anchored Rectangle Packing (CARP) problem, where rectangles are anchored in their center. We prove NP-hardness and provide a PTAS. In fact, our PTAS applies to any ARP problem where the anchor lies i...

Computational Geometry

We study several variations of line segment covering problem with axis-parallel unit squares in IR 2. A set S of n line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give constant factor approximation algorithms for those problems. For some variations, we have polynomial time exact algorithms. For the general version of the problem, where the segments are of arbitrary length and orientation, and the squares are given as input, we propose a factor 16 approximation result based on multilevel linear programming relaxation technique, which may be useful for solving some other problems. Further, we show that our problems have connections with the problems studied by Arkin et al. [2] on conflict-free covering problem. Our NP-completeness results hold for more simplified types of objects than those of Arkin et al. [2].

Journal of Experimental Algorithmics, 2007

We consider the problem of covering an orthogonal polygon with a minimum number of axis-parallel rectangles from a computational point of view. We propose an integer program which is the first general approach to obtain provably optimal solutions to this well-studied N P-hard problem. It applies to common variants like covering only the corners or the boundary of the polygon, and also to the weighted case. In experiments it turns out that the linear programming relaxation is extremely tight, and rounding a fractional solution is an immediate high quality heuristic. We obtain excellent experimental results for polygons originating from VLSI design, fax data sheets, black and white images, and for random instances. Making use of the dual linear program, we propose a stronger lower bound on the optimum, namely the cardinality of a fractional stable set. We outline ideas how to make use of this bound in primal-dual based algorithms. We give partial results which make us believe that our proposals have a strong potential to settle the main open problem in the area: To find a constant factor approximation algorithm for the rectangle cover problem.

Journal of Universal Computer Science

This paper discusses three rectilinear (that is, axis-parallel) covering problems in d dimensions and their variants. The first problem is the RECTILINEAR LINE COVER where the inputs are n points in ℝd and a positive integer k, and we are asked to answer if we can cover these n points with at most k lines where these lines are restricted to be axis parallel. We show that this problem has efficient fixed-parameter tractable (FPT) algorithms. The second problem is the RECTILINEAR k-LINKS SPANNING PATH PROBLEM where the inputs are also n points in ℝd and a positive integer k but here we are asked to answer if there is a piecewise linear path through these n points having at most k line-segments (links) where these line-segments are axisparallel. We prove that this second problem is FPT under the assumption that no two line-segments share the same line. The third problem is the RECTILINEAR HYPERPLANE COVER problem and we are asked to cover a set of n points in d dimensions with k axis-p...

Computational Geometry, 2011

We consider the Minimum Vertex Cover problem in intersection graphs of axisparallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families R where R 1 \ R 2 is connected for every pair of rectangles R 1 , R 2 ∈ R. This algorithm extends to intersection graphs of pseudodisks. The second algorithm achieves a factor of (1.5 + ε) in general rectangle families, for any fixed ε > 0, and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a non-trivial way.

Information Processing Letters, 1992

Algorithmica, 2002

In this paper we deal with two geometric problems arising from heterogeneous parallel computing: how to partition the unit square into p rectangles of given area s 1 , s 2 ,...,s p (such that p i=1 s i = 1), so as to minimize either (i) the sum of the p perimeters of the rectangles or (ii) the largest perimeter of the p rectangles? For both problems, we prove NP-completeness and we introduce a 7 4 -approximation algorithm for (i) and a (2/ √ 3)-approximation algorithm for (ii).

Theoretical Computer Science, 2020

We study a class of geometric covering and packing problems for bounded regions on the plane. We are given a set of axis-parallel line segments that induces a planar subdivision with a set of bounded (rectilinear) faces. We are interested in the following problems. (P1) Stabbing-Subdivision: Stab all bounded faces by selecting a minimum number of points in the plane. (P2) Independent-Subdivision: Select a maximum size collection of pairwise non-intersecting bounded faces. (P3) Dominating-Subdivision: Select a minimum size collection of faces such that any other face has a non-empty intersection (i.e., sharing an edge or a vertex) with some selected faces. We show that these problems are NP-hard. We even prove that these problems are NP-hard when we concentrate only on the rectangular faces of the subdivision. Further, we provide constant factor approximation algorithms for the Stabbing-Subdivision problem.

Algorithmica, 1997

The rectilinear polygon cover problem is one in which a certain class of features of a rectilinear polygon of n vertices has to be covered with the minimum number of rectangles included in the polygon. In particular, we consider covering the entire interior, the boundary and the set of corners of the polygon. These problems have important applications in storing images and in the manufacture of integrated circuits. Unfortunately, most of these problems are known to be NPcomplete. Hence it is necessary to develop e cient heuristics for these problems or to show that the design of e cient heuristics is impossible. In this paper we show: (a) The corner cover problem is NP-complete. (b) The boundary and the corner cover problem can be approximated within a ratio of 4 of the optimum in O(n log n) a n d O(n 1:5) time, respectively. (c) No polynomial-time approximation scheme exists for the interior and the boundary cover problems, unless P = NP.