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Titel:

Improved Approximation Algorithms for Box Contact Representations

AutorInnen:

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

Kategorie:

Artikel in Zeitschriften

erschienen in:

Algorithmica, Vol. 77, no. 3, 2017, pp. 902-920

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-hard on bipartite graphs of bounded maximum degree.

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BibTeX-Eintrag:

@Article{bdfkkpsw-iaabcr-a17, Title = {Improved Approximation Algorithms for Box Contact Representations}, Author = {Michael A. Bekos and Thomas C. van Dijk and Martin Fink and Philipp Kindermann and Stephen G. Kobourov and Sergey Pupyrev and Joachim Spoerhase and Alexander Wolff}, Journal = {Algorithmica}, Year = {2017}, Number = {3}, Pages = {902--920}, Volume = {77}, 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 \textsc{Contact Representation of Word Networks} (\textsc{Crown}) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. \textsc{Crown} is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, \textsc{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-hard on bipartite graphs of bounded maximum degree.}, Doi = {10.1007/s00453-016-0121-3} }