Realizability of Rectangular Euler Diagrams
Dominik Dürrschnabel, Uta Priss
TL;DR
The paper investigates the realizability of rectangular Euler diagrams by translating the problem into order-dimension properties of associated posets. It shows that 1D realizability corresponds to an Euler-poset of order-dimension $2$, enabling a polynomial-time construction, while 2D realizability corresponds to order-dimension $4$ and is NP-complete to decide, necessitating exponential-time methods in general. The authors connect their framework to prior work, notably Paetzold et al. (2023), and introduce extended Euler-posets to capture the 2D case, providing constructive criteria and algorithms. The results advance automatic generation of human-readable Euler diagrams from datasets and point to future work on heuristics and model relaxations.
Abstract
Euler diagrams are a tool for the graphical representation of set relations. Due to their simple way of visualizing elements in the sets by geometric containment, they are easily readable by an inexperienced reader. Euler diagrams where the sets are visualized as aligned rectangles are of special interest. In this work, we link the existence of such rectangular Euler diagrams to the order dimension of an associated order relation. For this, we consider Euler diagrams in one and two dimensions. In the one-dimensional case, this correspondence provides us with a polynomial-time algorithm to compute the Euler diagrams, while the two-dimensional case is linked to an NP-complete problem which we approach with an exponential-time algorithm.