Algorithmica

, Volume 77, Issue 3, pp 902–920

# Improved Approximation Algorithms for Box Contact Representations

• Michael A. Bekos
• Thomas C. van Dijk
• Martin Fink
• Philipp Kindermann
• Stephen Kobourov
• Sergey Pupyrev
• Joachim Spoerhase
• Alexander Wolff
Article

## Abstract

We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called Contact Representation of Word Networks (Crown) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Crown is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, Max-Crown, in which realizing each desired adjacency yields a certain profit. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-complete on bipartite graphs of bounded maximum degree.

### Keywords

Word clouds Box contact representations Approximation algorithms

## Notes

### Acknowledgments

We thank the anonymous reviewers for helping us to improve the presentation of our paper. We particularly thank the reviewer who contributed the idea to derandomize our algorithms for the general weighted case using Theorem 4.

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## Authors and Affiliations

• Michael A. Bekos
• 1
• Thomas C. van Dijk
• 2
• Martin Fink
• 2
• 3
• Philipp Kindermann
• 2
• Stephen Kobourov
• 4
• Sergey Pupyrev
• 4
• 5
• Joachim Spoerhase
• 2
• Alexander Wolff
• 2
1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany
2. 2.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
3. 3.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA
4. 4.Department of Computer ScienceUniversity of ArizonaTucsonUSA
5. 5.Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia