Compact Layered Drawings of General Directed Graphs

Abstract

Abstract. We consider the problem of layering general directed graphs under height and possibly also width constraints. Given a directed graph G = (V, A) and a maximal height, we propose a layering approach that minimizes a weighted sum of the number of reversed arcs, the arc lengths, and the width of the drawing. We call this the Compact Generalized Layering Problem (CGLP). Here, the width of a drawing is defined as the maximum sum of the number of vertices placed on a layer and the number of dummy vertices caused by arcs traversing the layer. The CGLP is NPhard. We present two MIP models for this problem. The first one (EXT) is our extension of a natural formulation for directed acyclic graphs as suggested by Healy and Nikolov. The second one (CGL) is a new formulation based on partial orderings. Our computational experiments on two benchmark sets show that the CGL formulation can be solved much faster than EXT using standard commercial MIP solvers. Moreover, we suggest a variant of CGL, called MML, that can be seen as a heuristic approach. In our experiments, MML clearly improves on CGL in terms of running time while it does not considerably increase the average arc lengths and widths of the layouts although it solves a slightly different problem where the dummy vertices are not taken into account.

A Natural Quadratic Approach to the Generalized Graph Layering Problem

“…The direct approach is a perfect fit for the DLP since the resulting problem can be solved in polynomial time by combinatorial algorithms as discussed by Gansner et al [2]. However, this property (more precisely, the underlying structure) is lost when incorporating width constraints or arc reversals, and the direct method becomes inferior to models with binary variables in practice [7,4].…”

“…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).…”

“…The aesthetic effects when integrating GLP-MS and GLP-W into the hierarchical framework by Sugiyama et al were extensively studied already in [4,6,7,8]. Here, we thus confine ourselves to evaluate (i) the computational effects when targeting different aesthetic objectives and aspect ratios, and (ii) the competitiveness of QLA-W and QLA-MS * with respect to CGL-W and CGL-MS * .…”

“…Also, we employ the same two instance sets as in the mentioned prior studies: The first set ATTar are the AT&T graphs from [1] whereof we extracted all non-tree instances with 20 ≤ |V | ≤ 60, and 20 ≤ |A| ≤ 168. Their density |A| |V | is within [1,4.72] (on average 1.47). The second set Random consists of 180 randomly generated and also acyclic and non-tree graphs with 17 to 60 vertices, and 30 to 91 arcs.…”

A Visualization Framework and User Studies for Overloaded Orthogonal Drawings

“…For a thorough treatment, we refer the reader to [HN13]. Additional references are [JMM*16, RESvH16]. A drawback of Sugiyama's approach is that it may create many edge crossings even if the graph is level‐planar (see, e.g.…”