TY - JOUR

T1 - Improved Approximation Algorithms for Box Contact Representations

AU - Bekos, Michael A.

AU - van Dijk, Thomas C.

AU - Fink, Martin

AU - Kindermann, Philipp

AU - Kobourov, Stephen

AU - Pupyrev, Sergey

AU - Spoerhase, Joachim

AU - Wolff, Alexander

N1 - Publisher Copyright:
© 2016, Springer Science+Business Media New York.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

KW - Approximation algorithms

KW - Box contact representations

KW - Word clouds

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U2 - 10.1007/s00453-016-0121-3

DO - 10.1007/s00453-016-0121-3

M3 - Article

AN - SCOPUS:84955618911

SN - 0178-4617

VL - 77

SP - 902

EP - 920

JO - Algorithmica

JF - Algorithmica

IS - 3

ER -