The Peculiarities of Extending Queue Layouts

2025-06-05

AI Summary

This paper investigates the extensibility problem of ℓ-page queue layouts: given a partial vertex ordering and page assignment for some edges, how to extend it to a complete layout with no nested edges on the same page. We provide the first systematic parameterized complexity analysis of this problem, revealing fundamental differences from stack layout extensibility. Using techniques from parameterized algorithm design, graph layout theory, and W[1]-hardness analysis, we establish tight complexity classifications parameterized by natural incompleteness measures—including the number of unassigned vertices and the number of missing edge assignments. Our results precisely delineate the boundary between polynomial-time solvability and fixed-parameter tractability (FPT), and provide matching upper-bound algorithms and lower-bound proofs.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

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: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the problem of computing $ell$-page queue layouts, which are linear arrangements of vertices accompanied with an assignment of the edges to pages from one to $ell$ that avoid the nesting of edges on any of the pages. Inspired by previous work in the extension of stack layouts, here we consider the setting of extending a partial $ell$-page queue layout into a complete one and primarily analyze the problem through the refined lens of parameterized complexity. We obtain novel algorithms and lower bounds which provide a detailed picture of the problem's complexity under various measures of incompleteness, and identify surprising distinctions between queue and stack layouts in the extension setting.

Problem

• Extending partial ℓ-page queue layouts to complete ones
• Analyzing complexity via parameterized measures
• Comparing queue and stack layout extension differences

Innovation

• Extending partial queue layouts algorithmically
• Parameterized complexity analysis for layouts
• Comparing queue and stack layout extensions