Core Concepts
Parameterized complexity analysis of Constrained and Ordered Level Planarity.
Abstract
The content discusses the parameterized complexity of Constrained and Ordered Level Planarity, focusing on height as a parameter. It introduces the problems CLP and OLP, their variants, and previous research results. The article presents a reduction from Multicolored Independent Set to Ordered Level Planarity, proving W[1]-hardness. Additionally, it extends this construction to show XNLP-hardness using a reduction from Chained Multicolored Independent Set. The tractability of 3-level CLP is outlined with assumptions and constraints propagation explained.
Introduction to Level Planarity problems: CLP, OLP, variants.
Reductions from MCIS and CMCIS to prove hardness.
Tractability analysis for 3-level CLP with assumptions.
Stats
Previous results by Brückner and Rutter [SODA 2017] state that both CLP and OLP are NP-hard even in severely restricted cases.
Klemz and Rote [24] showed that OLP (and thus CLP) is NP-hard even when restricted to the case where the underlying undirected graph of G is a disjoint union of paths.
Brückner and Rutter [7] provided a reduction showing that CLP is NP-hard; their reduction shows the NP-hardness of Partial Level Planarity, which can be seen as a generalization of OLP and a special case of CLP.
Bodlaender et al. introduced XNLP-completeness for several problems parameterized by linear width measures such as Capacitated Dominating Set by pathwidth and Max Cut by linear cliquewidth.
Quotes
"Ordered Level Planarity parameterized by the height of the input graph is XNLP-complete." - Theorem 1
"Constrained Level Planarity is NP-hard even when restricted to height 4." - Theorem 2
"An XP / XNLP Algorithm for Ordered Level Planarity." - Section 2 title