On Layered Area-Proportional Rectangle Contact Representations

TL;DR

The paper studies layered word-cloud representations where words are axis-aligned rectangles of equal height placed on $L$ layers, formalizing two variants: Max-Layered-Crown (maximizing contacts with no false adjacencies) and Max-Int-Layered-Crown (with integer widths and coordinates). It proves NP-completeness for k-Int-Layered-Crown on internally triangulated graphs and for k-Layered-Crown on planar graphs, using intricate gadgets that encode variables, clauses, and propagation of assignments. It then delivers algorithmic results: a 1/2-approximation for Max-Layered-Crown on triangulated graphs, an XP dynamic-programming algorithm for Max-Int-Layered-Crown, and a Baker-based PTAS for Max-Int-Layered-Crown when the rectangle width is polynomial in the input size. These results advance understanding of the computational complexity and tractability of structured word-cloud representations and provide practical algorithms for relevant instances.

Abstract

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 $n$.

Figures (11)

• Figure 1: Randomly arranged word cloud (left) and semantic word cloud (right), generated using the first chapter of "Alice's Adventures in Wonderland" by Lewis Carroll.
• Figure 2: Internally triangulated graph with 3 layers (left) and a contact representation (right).
• Figure 3: (a) Contact representation and (b) underlying graph for a variable gadget with variable assignment true; (c) variable gadget with variable assignment false; (d,e) alternative representations with the same number of realized contacts; (f) valid representation realizing fewer contacts.
• Figure 4: (a) Moving wall vertices leads to false adjacencies (red curve) and (b) propagating variable values.
• Figure 5: Contact representation (top) and underlying graph (bottom) for a clause gadget, including multiple examples of placements for $v_s$. Unrealized edges between $v_s$ and vertices of adjacent layers are omitted for readability.

Theorems & Definitions (10)

• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• theorem thmcountertheorem: thm:lp-crown-np
• lemma thmcounterlemma: Noellenburg:21, Theorem 2
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof
• lemma thmcounterlemma