Packing squares into a square

Leung, Joseph Y.-T.; Tam, Tommy W.; Wong, C. S.; Young, Gilbert H.; Chin, F.

doi:10.1016/0743-7315(90)90019-l

Cited by 132 publications

(54 citation statements)

References 2 publications

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“…The problem is known to be NP-complete even if the box is a square, that is, if W = H [16]. The BOX PACKING problem is equivalent to AREA-CROWN when the profit graph has no edges.…”

Section: The Area-crown Problemmentioning

confidence: 99%

Semantic Word Cloud Representations: Hardness and Approximation Algorithms

Barth

Fabrikant

Kobourov

et al. 2014

LATIN 2014: Theoretical Informatics

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Abstract. We study a geometric representation problem, where we are given a set B of axis-aligned rectangles (boxes) with fixed dimensions and a graph with vertex set B. The task is to place the rectangles without overlap such that two rectangles touch if the graph contains an edge between them. We call this problem CONTACT REPRESENTATION OF WORD NETWORKS (CROWN). It formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Here, we represent words by rectangles and semantic relationships by edges. We show that CROWN is strongly NP-hard even if restricted to trees and weakly NP-hard if restricted to stars. We also consider the optimization problem MAX-CROWN where each adjacency induces a certain profit and the task is to maximize the sum of the profits. For this problem, we present constant-factor approximations for several graph classes, namely stars, trees, planar graphs, and graphs of bounded degree. Finally, we evaluate the algorithms experimentally and show that our best method improves upon the best existing heuristic by 45%.

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“…The problem is known to be NP-complete even if the box is a square, that is, if W = H [16]. The BOX PACKING problem is equivalent to AREA-CROWN when the profit graph has no edges.…”

Section: The Area-crown Problemmentioning

confidence: 99%

Semantic Word Cloud Representations: Hardness and Approximation Algorithms

Barth

Fabrikant

Kobourov

et al. 2014

LATIN 2014: Theoretical Informatics

View full text Add to dashboard Cite

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“…In recent years, there has been increasing interest in extensions of packing problems such as strip packing [1,2,14,17,19], knapsack [3,15,18] and bin packing [4,5,6,9,20], to multiple independent criteria (vector packing) or multiple dimensions (geometric packing).…”

Section: Introductionmentioning

confidence: 99%

Two for One: Tight Approximation of 2D Bin Packing

Jansen

Prädel

Schwarz

2009

Lecture Notes in Computer Science

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In this paper, we study the two-dimensional geometrical bin packing problem (2DBP): given a list of rectangles, provide a packing of all these into the smallest possible number of unit bins without rotating the rectangles. Beyond its theoretical appeal, this problem has many practical applications, for example in print layout and VLSI chip design.We present a 2-approximate algorithm, which improves over the previous best known ratio of 3, matches the best results for the problem where rotations are allowed and also matches the known lower bound of approximability. Our approach makes strong use of a PTAS for a related 2D knapsack problem and a new algorithm that can pack instances into two bins if OPT = 1.

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“…This means that the one-dimensional problem cannot be approximated up to a factor smaller than 3 2 , unless P = N P , (due to a simple reduction from the PARTITION problem, see problem SP12 in [10]). Also, the two-dimensional problem cannot be approximated up to a factor smaller than 2, unless P = N P , since it was shown in [20] that given a set of squares, it is N P -hard to check whether these squares can be packed into one bin. These results hold for the graph classes we consider since an empty graph (i.e., a graph with an empty edge set) is both bipartite and perfect.…”

Section: Introductionmentioning

confidence: 99%

Multi-dimensional Packing with Conflicts

Epstein

Levin

Stee³

2007

Fundamentals of Computation Theory

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We study the two-dimensional version of the bin packing problem with conflicts. We are given a set of (two-dimensional) squares V = {1, 2, . . . , n} with sides s 1 , s 2 . . . , s n ∈ [0, 1] and a conflict graph G = (V, E). We seek to find a partition of the items into independent sets of G, where each independent set can be packed into a unit square bin, such that no two squares packed together in one bin overlap. The goal is to minimize the number of independent sets in the partition.This problem generalizes the square packing problem (in which we have E = ∅) and the graph coloring problem (in which s i = 0 for all i = 1, 2, . . . , n). It is well known that coloring problems on general graphs are hard to approximate. Following previous work on the one-dimensional problem, we study the problem on specific graph classes, namely, bipartite graphs and perfect graphs.We design a 2+ε-approximation for bipartite graphs, which is almost best possible (unless P = N P ). For perfect graphs, we design a 3.2744-approximation.

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Packing squares into a square

Cited by 132 publications

References 2 publications

Semantic Word Cloud Representations: Hardness and Approximation Algorithms

Semantic Word Cloud Representations: Hardness and Approximation Algorithms

Two for One: Tight Approximation of 2D Bin Packing

Multi-dimensional Packing with Conflicts

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