Minimum-Width Drawing of Trees with Sized Vertices

2026-07-29

AI Summary

This study addresses the minimum-width layered layout problem for trees with fixed-size rectangular nodes, where child node ordering may be reordered to minimize overall width. We prove that this problem remains NP-complete even for unit-width binary trees. To tackle it, we propose a novel hybrid framework that integrates exact mixed-integer linear programming (MILP) formulations with an efficient heuristic algorithm. Experimental results demonstrate that allowing reordering of children reduces layout width by approximately 20% on average, with improvements reaching up to 55% in some cases. The proposed heuristic computes layouts for large-scale instances within one second and yields solutions within 1.25 times the optimal width in at least 75% of tested cases.

Abstract

Trees arise in many applications and computing nice tree layouts is a classical problem in information visualization. In many practical settings, vertices need to be represented as rectangles with a given width and height rather than as points. When an order over the children of each vertex is given, polynomial-time algorithms are known that produce drawings adhering to various drawing conventions. However, in many applications, the order of children carries no semantic meaning, and choosing it well can significantly reduce the drawing's width. In this paper, we study the problem \textsc{Min-Width Tree Drawing with Reordering} (\textsc{MWD}): given a rooted tree whose vertices have prescribed widths, find a sibling order at each internal vertex that minimizes the width of the resulting layered drawing. We show that the problem is \textsf{NP}-complete, even on binary trees with unit-width vertices. We present a mixed integer linear program that solves \textsc{MWD} exactly on moderately sized instances, and a heuristic that is fast and delivers good results in practice. We evaluate both approaches against a baseline on synthetic and real-world datasets, where reordering reduces drawing width by a median of $\approx20\%$ and by up to $\approx55\%$ on individual instances. The heuristic computes its layouts in under a second and, when the MILP proves optimality, it stays within $25\%$ of the optimal width in three-quarters of all instances.

Problem

• tree drawing
• minimum width
• vertex reordering
• layered layout
• NP-complete

Innovation

• tree drawing
• minimum width
• sibling reordering
• mixed integer linear programming
• NP-completeness