Minimum Number Of Rectangles Research Articles

Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition

Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig--Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a "price-and-verify" algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature.

Delta DLP 3-D Printing of Large Models

This paper presents a 3-D printing system that uses a low-cost off-the-shelf consumer projector to fabricate large models. Compared with traditional digital light processing (DLP) 3-D printers using a single vertical carriage, the platform of our DLP 3-D printer using delta mechanism can also move horizontally in the plane. We show that this system can print 3-D models much larger than traditional DLP 3-D printers. The major challenge to realize 3-D printing of large models in our system comes from how to cover a planar polygonal domain by a minimum number of rectangles with fixed size, which is NP-hard. We propose a simple yet efficient approximation algorithm to solve this problem. The key idea is to segment a polygonal domain using its medial axis and afterward merge small parts in the segmentation. Given an arbitrary polygon $\Omega $ with $n$ generators (i.e., line segments and reflex vertices in $\Omega $ ), we show that the time complexity of our algorithm is $O(n^{2}\log ^{2}~n)$ and the number of output rectangles covering $\Omega $ is $O(Kn)$ , where $K$ is an input-polygon-dependent constant. A physical prototype system is built and several large 3-D models with complex geometric structures have been printed as examples to demonstrate the effectiveness of our approach. Note to Practitioners —Low-cost 3-D printers and 3-D printing of large models are two important but often conflicting goals in manufacturing industry. Usually low-cost devices such as DLP 3-D printer using an off-the-shelf consumer projector cannot print large models, due to the small projection area of the projector. In this paper, we propose to horizontally move the platform such that a large area can be printed by the composition of multiple small projection areas. Based on this new mechanism, we propose a simple yet efficient algorithm to cover an arbitrary polygonal shape (possibly with holes or multiple disjoint polygons) by a small set of rectangles with fixed size. Our algorithm is theoretically sound and can be easily implemented. We built a physical prototype system of the proposed low-cost Delta DLP 3-D printer, which successfully prints several large models with satisfied mechanical properties.

Covering Orthogonal Polygons with Non-Piercing Rectangles

Given a simply connected orthogonal polygon P, a polynomial time algorithm is presented to cover the polygon with the minimum number of rectangles, under the restriction that if A and B are two overlapping rectangles in the cover then either A - B or B - A is connected. The algorithm runs in O(n log n + nm) time, where n is the number of vertices of P and m is the number of edges in the visibility graph of P that are either horizontal, vertical or form the diagonal of an empty rectangle.

Minimum dissection of a rectilinear polygon with arbitrary holes into rectangles

In this paper, the problem of dissecting a plane rectilinear polygon with arbitrary (possibly, degenerate) holes into a minimum number of rectangles is shown to be solvable inO(n3/2 logn) time. This fact disproves a famous assertion about the NP-hardness of the minimum rectangular dissection problem for rectilinear polygons with point holes.

COVERING CONVEX RECTILINEAR POLYGONS IN LINEAR TIME

In this paper, we discuss the problem of covering a convex rectilinear polygon with the minimum number of rectangles. The rectangles can overlap with each other. We show that this problem can be solved in linear time, which improves the previous O(n2) result proposed by Franzblau and Kleitman3. However, Franzblau and Kleitman's algorithm works for a larger class of polygons (vertically convex).

Computational-geometric methods for polygonal approximations of a curve

In cartography, computer graphics, pattern recognition, etc., we often encounter the problem of approximating a given finer piecewise linear curve by another coarser piecewise linear curve consisting of fewer line segments. In connection with this problem, a number of papers have been published, but it seems that the problem itself has not been well modelled from the standpoint of specific applications, nor has a nice algorithm, nice from the computational-geometric viewpoint, been proposed. In the present paper, we first consider (i) the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the former, and show that an optimum solution of this problem can be found in a polynomial time. We also mention recent results on related problems by several researchers including the authors themselves. We then pose (ii) a problem of covering a sequence of n points by a minimum number of rectangles with a given width, and present an O(n long n)-time algorithm by making use of some fundamental established techniques in computational geometry. Furthermore, an O(mn(logn)2)-time algorithm is presented for finding the minimum width w such that a sequence of n points can be covered by at most m rectangles with width w. Finally, (iii) several related problems are discussed.

An algorithm for covering polygons with rectangles

Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is NP -hard. However, we give here an O ( v 2 ) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here v is the number of vertices.) The problem is first reduced to a 1-dimensional interval “basis” problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E. Györi ( J. Combin Theory Ser. B 37 , No. 1, 1–9).

It’s Hard to Color Antirectangles

The maximum number of elements in an antirectangle of a convex board equals the minimum number of rectangles it takes to cover that board. It is shown here that the dual of this theorem, that the minimum number of antirectangles needed to cover a convex board equals the maximum size of any rectangle of the board, is not generally true.

Covering Regions by Rectangles

A board$\mathcal{B}$ is a finite set of unit squares lying in the plane whose corners have integer coordinates. A rectangle of $\mathcal{B}$ is a rectangular subset of $\mathcal{B}$ and an antirectangle is a set of squares in $\mathcal{B}$ no two of which are in a common rectangle. We prove a conjecture of Chvátal that $\mathcal{B}$ if is convex in the horizontal and vertical directions, then the minimum number of rectangles whose union is $\mathcal{B}$ equals the maximum cardinality of an antirectangle. Our proof uses two analogous minimax theorems about covering the corners and covering the edges of the board. We quote examples that illustrate the necessity of the hypotheses, and give some conjectures and open questions. The method of proof can give a polynomial running time algorithm for finding a minimum cover.