Segment Intersection Representations, Level Planarity and Constrained Ordering Problems

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F26%3A10515269" target="_blank" >RIV/00216208:11320/26:10515269 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/978-3-032-11835-6_15" target="_blank" >https://doi.org/10.1007/978-3-032-11835-6_15</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/978-3-032-11835-6_15" target="_blank" >10.1007/978-3-032-11835-6_15</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Segment Intersection Representations, Level Planarity and Constrained Ordering Problems

Popis výsledku v původním jazyce

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 THERE EXISTS R-complete [21]) in the general case [15] and remains so if we restrict the segments to be axis-aligned, i.e., horizontal and vertical [14]. 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 [14, 16]. We resolve this question by giving efficient quartic-time 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 if all edges connect adjacent levels. 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.

Název v anglickém jazyce

Segment Intersection Representations, Level Planarity and Constrained Ordering Problems

Popis výsledku anglicky

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 THERE EXISTS R-complete [21]) in the general case [15] and remains so if we restrict the segments to be axis-aligned, i.e., horizontal and vertical [14]. 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 [14, 16]. We resolve this question by giving efficient quartic-time 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 if all edges connect adjacent levels. 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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GX23-04949X" target="_blank" >GX23-04949X: Stěžejní otázky diskrétní geometrie</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2026

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Lecture Notes in Computer Science

ISBN

978-3-032-11834-9

ISSN

—

e-ISSN

1611-3349

Počet stran výsledku

14

Strana od-do

205-218

Název nakladatele

Springer Nature

Místo vydání

Cham

Místo konání akce

Graph-Theoretic Concepts in Computer Science

Datum konání akce

11. 6. 2025

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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