Rectangle tiling np hard

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August undefined, 2024

Webany rectangle (the weight of a rectangle is the sum of elements which are covered by it). We prove that it is NP-hard to approximate this problem to within a factor of 11 3 (the previous best result was 11 4). I. Introduction RTILE problem. Given an n×n array A of positive numbers, ﬁnd a tiling using at most p rectangles (that is rectangles ... WebJan 11, 2015 · The Square Tiling Problem was recently introduced as equivalent to the problem of reconstructing an image from patches and a possible general-purpose indexing tool. Unfortunately, the Square Tiling Problem was shown to be NP-hard. A 1/2-approximation is known.

Tile-Packing Tomography Is NP-hard

WebMar 4, 2024 · We prove that it is NP-hard to approximate this problem to within a factor of \textbf{1$\frac{1}{3}$} (the previous best result was $1\frac{1}{4}$). Discover the world's research 20+ million members WebJul 28, 2024 · The problem of rectangle tiling binary arrays is deﬁned as follows. Given an n×narray Aof zeros and ones and a natural number p, our task is to partition Ainto at most prectangular tiles, so... it might have been poem analysis

Improved Approximation Algorithms for Rectangle Tiling and …

WebRectangle Tiling Binary Arrays Pratik Ghosal 1, Syed Mohammad Meesum2, and Katarzyna Paluch 1 University of Wrocław, Wrocław, Poland 2 KREA University, Sricity, India ... Both the RTILE and DRTILE problems have been proven to be NP-hard [7]. Grigni and Manne [6] proved that optimal p ptiling (which is a restricted variant of the RTILE Web(its width or height is 1), while for some other types of tiles NP-hardness results have been shown in the literature. In this paper we present a complete solution to this question by showing that the problem remains NP-hard for all tiles other than bars. Keywords Tilings ·Discrete tomography ·NP-hardness ·Afﬁne independence M. Chrobak WebNov 1, 2001 · We initiate a parameterized complexity study of the NP-hard problem to tile a positive integer matrix with rectangles, keeping the number of tiles and their maximum … neil young sugar mountain farm aid

Is packing rectangles exactly into a larger rectangle NP …

WebNov 1, 2000 · Aspects of a multivariate complexity analysis for Rectangle Tiling. 2011, Operations Research Letters. Show abstract. We initiate a parameterized complexity study of the NP-hard problem to tile a positive integer matrix with rectangles, keeping the number of tiles and their maximum weight small. We show that the problem remains NP-hard … WebWe show that Rectangle Tiling remains NP-hard when the input matrix is binary, w= 1, and the rectangle shapes are only from {1 × 1, 2 × 2}. In particular, this implies that tiling with squares, in the following called Square Tiling, is NP-hard. On the positive side, we observe that RectangleTilingcan be solved in O((mn)plogp) time and, for neil young stars and barshttp://fpt.akt.tu-berlin.de/publications/tiling.pdf it might help clear things up

"WebMar 2, 2003 · The RTile problem consists of finding the minimum number of rectangles that cover a given matrix if the sum of of the entries within each tiles is bounded by a given constant value. This problem... " - Rectangle tiling np hard

Rectangle tiling np hard

Efficient Approximation Algorithms for Tiling and Packing …

WebRectangle Tiling remains NP-hard for binary matrices tiled with 1 1- or 2 2-squares and restricting the maxi-mum weight to 1. To this end, we devise a reduction from the NP-hard … WebOn the other hand, we provide a near-linear time algorithm that returns a solution at most 2.5 times the optimal. Other rectangle tiling and packing problems that we study have similar properties: while it is easy to solve them optimally in one dimension, the two-dimensional versions become NP-hard.

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WebJan 1, 2004 · Rectangle Tiling is NP-hard [11]. Several polynomial-time constant-factor approximation algorithms have been developed for its two corresponding optimization … WebOther rectangle tiling and packing problems that we study have similar properties: while it is easy to solve them optimally in one dimension, the two-dimensional versions become NP …

WebWe prove that in the twodimensional case it is NP-hard to approximate this problem to within a factor of 1:25. On the other hand, we provide a near-linear time algorithm that returns a solution at most 2:5 times the optimal. Other rectangle tiling and p... WebSep 1, 2011 · We extend their result by showing that Rectangle Tiling remains NP-hard for binary matrices tiled with 1×1- or 2×2-squares and restricting the maximum weight to 1. To this end, we devise a reduction from the NP-hard Planar Vertex Cover problem . Theorem 2.1. Square Tiling is NP-hard on binary matrices and w = 1.

http://akt.tu-berlin.de/fileadmin/fg34/publications-akt/tiling.pdf WebMay 23, 2024 · I find it hard to construct a reduction from the 2D-bin-packing problem to this problem, since this very special, restricted case of fitting the rectangles exactly might be …

WebMay 12, 2024 · In lecture2, it is stated that Rectangle Packing is NP-hard. I can understand this because the problem can be reduced to 3-partition problem But I don't know why it's …

Webof a tiling. Instead, let us use four colors, as shown above. Any 1×4 tile that we place on this board will cover an even number (possi-bly zero) of squares of each color. Therefore, if we had a tiling of the board, the total number of squares of each color would be even. But there are 25 squares of each color, so a tiling is impossible. it might have the shakes crosswordWebIf the array A were one-dimensional, this problem could be easily solved by dynamic programming. We prove that in the twodimensional case it is NP-hard to approximate this problem to within a factor of 1:25. On the other hand, we provide a near-linear time algorithm that returns a solution at most 2:5 times the optimal. Other rectangle tiling... neil young t-bone lyricsWebMar 4, 2024 · Abstract: The problem that we consider is the following: given an $n \times n$ array $A$ of positive numbers, find a tiling using at most $p$ rectangles (which means … it might go without sayingWebNov 1, 2001 · Rectangle Tiling is NP-hard [11]. Several polynomial-time constant-factor approximation algorithms have been developed for its two corresponding optimization … neil young tell me whyの和訳Webby some rectangle and no two rectangles must overlap) that minimizes the maximum weight of any rectangle (the weight of a rectangle is the sum of elements which are … it might help clear things up crosswordWebPut the largest rectangle remaining into your packed area. If it can't fit anywhere, place it in a place that extends the pack region as little as possible. Repeat until you finish with the smallest rectangle. It's not perfect at all but it's easy and a nice baseline. it might have been cole razzanoWebSince it is NP-hard to even ﬁnd a feasible solution for this dual problem [FPT81], it cannot be approximated to within any factor. Hence, we do not consider this dual problem any further. 1.2 Motivating Applications Rectangle tiling and packing problems as deﬁned aboveare natural combina-torial problems arising in many scenarios. For motiva- it might have been moved renamed or deleted