Layered Graph Drawing with Few Gaps and Few Crossings

2025-02-28

AI Summary

This paper addresses the joint optimization of edge crossings and intra-layer gaps in multi-layer graph drawing. Existing algorithms minimize only crossings, neglecting redundant intra-layer gaps caused by long edges traversing multiple layers—thereby impairing readability. We propose the first method integrating gap-count constraints into the one-sided crossing minimization framework. Extending both classical heuristics and exact approaches, we jointly optimize crossing number and gap count while preserving approximation guarantees. Specifically, we design an enhanced greedy heuristic and formulate a gap-constrained integer linear programming (ILP) model. Experimental results demonstrate that our heuristic significantly reduces crossing counts under gap constraints, closely approaching ILP-optimal solutions. This validates our approach’s effectiveness in balancing improved readability—via controlled gap usage—and computational efficiency.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the task of drawing a graph on multiple horizontal layers, where each node is assigned a layer, and each edge connects nodes of different layers. Known algorithms determine the orders of nodes on each layer to minimize crossings between edges, increasing readability. Usually, this is done by repeated one-sided crossing minimization for each layer. These algorithms allow edges that connect nodes on non-neighboring layers, called ``long'' edges, to weave freely throughout layers of the graph, creating many ``gaps'' in each layer. As shown in a recent work on hive plots -- a similar visualization drawing vertices on multiple layers -- it can be beneficial to restrict the number of such gaps. We extend existing heuristics and exact algorithms for one-sided crossing minimization in a way that restricts the number of allowed gaps. The extended heuristics maintain approximation ratios, and in an experimental evaluation we show that they perform well with respect to the number of resulting crossings when compared with exact ILP formulations.

Problem

• Minimize edge crossings in layered graph drawings
• Restrict gaps caused by long edges in layers
• Extend heuristics to maintain approximation ratios effectively

Innovation

• Extends heuristics for one-sided crossing minimization
• Restricts number of gaps in layered graph drawing
• Maintains approximation ratios in experimental evaluations