Parameterized Approaches to Orthogonal Compaction

TL;DR

This paper studies the orthogonal compaction problem for planar graph drawings, showing it is fixed-parameter tractable when parameterized by the number of kitty corners, which are pairs of reflex corners pointing at each other.

Contribution

It introduces the parameter of kitty corners and proves that orthogonal compaction is fixed-parameter tractable with respect to this parameter.

Findings

Orthogonal compaction is NP-hard in general.

The problem is fixed-parameter tractable when parameterized by kitty corners.

Presence of kitty corners influences the problem's complexity.

Abstract

Orthogonal graph drawings are used in applications such as UML diagrams, VLSI layout, cable plans, and metro maps. We focus on drawing planar graphs and assume that we are given an \emph{orthogonal representation} that describes the desired shape, but not the exact coordinates of a drawing. Our aim is to compute an orthogonal drawing on the grid that has minimum area among all grid drawings that adhere to the given orthogonal representation. This problem is called orthogonal compaction (OC) and is known to be NP-hard, even for orthogonal representations of cycles [Evans et al., 2022]. We investigate the complexity of OC with respect to several parameters. Among others, we show that OC is fixed-parameter tractable with respect to the most natural of these parameters, namely, the number of \emph{kitty corners} of the orthogonal representation: the presence of pairs of kitty corners in an orthogonal representation makes the OC problem hard. Informally speaking, a pair of kitty corners is a pair of reflex corners of a face that point at each other. Accordingly, the number of kitty corners is the number of corners that are involved in some pair of kitty corners.