Improved Approximation Algorithms for Box Contact Representations

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• 作者：Michael A. Bekos ; Thomas C. van Dijk ; Martin Fink ; Philipp Kindermann…
• 关键词：Word clouds ; Box contact representations ; Approximation algorithms
• 刊名：Algorithmica
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：77
• 期：3
• 页码：902-920
• 全文大小：
• 刊物类别：Computer Science
• 刊物主题：Algorithm Analysis and Problem Complexity; Theory of Computation; Mathematics of Computing; Algorithms; Computer Systems Organization and Communication Networks; Data Structures, Cryptology and Inform
• 出版者：Springer US
• ISSN：1432-0541
• 卷排序：77

文摘

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.