Compact Layered Drawings of General Directed Graphs

TL;DR

This paper introduces the Compact Generalized Layering Problem (CGLP) for layered graph drawings, proposing two MIP models and a heuristic, to optimize layout quality under height and width constraints, addressing NP-hardness.

Contribution

It presents two novel MIP formulations for CGLP and a heuristic approach, improving computational efficiency and layout quality in layered graph visualization.

Findings

CGL formulation is faster than EXT with standard solvers.

MML heuristic reduces runtime significantly.

MML maintains similar arc lengths and widths as exact methods.

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 NP-hard. 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.