On Layered Area-Proportional Rectangle Contact Representations

Semantic word clouds visualize the semantic relatedness between the words of a text by placing pairs of related words close to each other. Formally, the problem of drawing semantic word clouds corresponds to drawing a rectangle contact representation of a graph whose vertices correlate to the words to be displayed and whose edges indicate that two words are semantically related. The goal is to maximize the number of realized contacts while avoiding any false adjacencies. We consider a variant of this problem that restricts input graphs to be layered and all rectangles to be of equal height, called \textsc{Maximum Layered Contact Representation Of Word Networks} or \textsc{Max-LayeredCrown}, as well as the variant \textsc{Max-IntLayeredCrown}, which restricts the problem to only rectangles of integer width and the placement of those rectangles to integer coordinates. We classify the corresponding decision problem $k$-\textsc{IntLayeredCrown} as NP-complete even for triangulated graphs and $k$-\textsc{LayeredCrown} as NP-complete for planar graphs. We introduce three algorithms: a 1/2-approximation for \textsc{Max-LayeredCrown} of triangulated graphs, and a PTAS and an XP algorithm for \textsc{Max-IntLayeredCrown} with rectangle width polynomial in $n$.