Segment Intersection Representations, Level Planarity and Constrained Ordering Problems
Simon D. Fink, Matthias Pfretzschner, Peter Stumpf
TL;DR
The paper resolves the open question of tractability for segment intersection representations when one axis order is fixed by presenting two complementary approaches. A graph-drawing route reduces SF-HV-SEG to a proper $\mathcal{T}$-Level Planarity instance, yielding a quadratic-time solution and an overall $O(n^4)$ algorithm, while a purely combinatorial route encodes the problem as a constrained ordering task via PQ-trees and Sequential PQ-Ordering, achieving a direct quadratic-time solution. The authors prove a tight quadratic-time equivalence between SF-HV-SEG and proper $\mathcal{T}$-Level Planarity and show how to transform instances into matchings with a single per-level constraint, further strengthening tractability. They also connect these problems to 2-fixed Simultaneous PQ-Ordering, enabling a unified, efficient framework across geometric, graph-drawing, and combinatorial perspectives, effectively unifying six equivalent formulations. This work closes a 30-year open problem and links the study to related topics such as $k$-ary Tangles, with potential implications for constrained orderings and planar representations.
Abstract
In the Segment Intersection Graph Representation Problem, we want to represent the vertices of a graph as straight line segments in the plane such that two segments cross if and only if there is an edge between the corresponding vertices. This problem is NP-hard (even $\exists\mathbb{R}$-complete [Schaefer, 2010]) in the general case [Kratochvíl & Neŝetril, 1992] and remains so if we restrict the segments to be axis-aligned, i.e., horizontal and vertical [Kratochvíl, 1994]. A long standing open question for the latter variant is its complexity when the order of segments along one axis (say the vertical order of horizontal segments) is already given [Kratochvíl & Neŝetril, 1992; Kratochvíl, 1994]. We resolve this question by giving efficient solutions using two very different approaches that are interesting on their own. First, using a graph-drawing perspective, we relate the problem to a variant of the well-known Level Planarity problem, where vertices have to lie on pre-assigned horizontal levels. In our case, each level also carries consecutivity constraints on its vertices; this Level Planarity variant is known to have a quadratic solution. Second, we use an entirely combinatorial approach, and show that both problems can equivalently be formulated as a linear ordering problem subject to certain consecutivity constraints. While the complexity of such problems varies greatly, we show that in this case the constraints are well-structured in a way that allows a direct quadratic solution. Thus, we obtain three different-but-equivalent perspectives on this problem: the initial geometric one, one from planar graph drawing and a purely combinatorial one.