Generalized Layerings for Arbitrary and Fixed Drawing Areas

Abstract

The Directed Layering Problem (DLP) solves a step of the widely used layer-based approach to automatically draw directed acyclic graphs. To cater for cyclic graphs, usually a preprocessing step is used that solves the Feedback Arc Set Problem (FASP) to make the graph acyclic before a layering is determined. Here we present the Generalized Layering Problem (GLP), which solves the combination of DLP and FASP simultaneously, allowing general graphs as input. We present an integer programming model and a heuristic to solve the NP-complete GLP and perform thorough evaluations on different sets of graphs and with different implementations for the steps of the layer-based approach. We observe that GLP reduces the number of dummy nodes significantly, can produce more compact drawings, and improves on graphs where DLP yields poor aspect ratios. The drawings resulting from GLP also turn out to be more suitable for making the best possible use of a given drawing area. However, we show that a specialized variant of GLP can yield considerable improvements w.r.t. this particular optimization goal.

A Natural Quadratic Approach to the Generalized Graph Layering Problem

“…In terms of the latter, the ordering-based CGL models (a compacted reformulation of those in [4] and [8] is displayed in the appendix) are more economical. Even if auxiliary variables for arc reversals or dummy vertices are introduced, their total number is still only (|V | + |A|) · (Y − 1).…”

“…In this experiment, the parameters are set (almost) as in [8]: Y = |V |, ω len = 1, ω rev = Y ω len |A|, and ω scl = ω rev |A| + |A| · Y + 1. Priorities are thus the same as in experiment (2) before, except now emphasizing on a maximum scaling factor instead of the width alone.…”

“…This motivates the recently studied integration of the first two steps [4,6,7,8]. Here, the central idea is to identify (a small number of) suitable arcs to be drawn reverse to the intended hierarchical direction such that this enables a layout that is two-dimensionally compact, possibly meets other common aesthetic criteria such as a minimum total arc length, or even adapts to a drawing area of a certain aspect ratio.…”

“…Finally, Rüegg et al propose to optimize a layering with respect to a target drawing area of width r W and height r H [8]. Informally, a 'best' such drawing is considered one that can be maximally scaled ('zoomed in') until it exhausts one of the two dimensions (cf.…”

“…A slight variation of GLP-MS as well proposed in [8] will be denoted GLP-MS * : Here, instead of minimizing −ω scl S, one minimizes ω scl S where S := 1 S .…”