Linear Layouts of Graphs with Priority Queues

2025-06-30

AI Summary

This paper introduces the “priority queue layout” model for edge-weighted graphs: given a linear ordering of vertices, edges are assigned to multiple pages (priority queues), where edges in each page are processed in non-decreasing order of weight, and edges across different pages must not be nested. The contributions are threefold: (1) formalizing the model for the first time, characterizing single-page layouts as exactly those graphs with pathwidth one, and providing a linear-time recognition algorithm; (2) establishing that some edge-weighted graphs require Ω(n) pages, and proving that the minimum number of pages needed lies between the graph’s pathwidth and treewidth; (3) proving that computing the minimum number of pages under a fixed vertex ordering is NP-complete. The work integrates graph theory, combinatorial optimization, and computational complexity analysis, establishing a novel paradigm and theoretical foundation for linear layouts of weighted graphs.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A linear layout of a graph consists of a linear ordering of its vertices and a partition of its edges into pages such that the edges assigned to the same page obey some constraint. The two most prominent and widely studied types of linear layouts are stack and queue layouts, in which any two edges assigned to the same page are forbidden to cross and nest, respectively. The names of these two layouts derive from the fact that, when parsing the graph according to the linear vertex ordering, the edges in a single page can be stored using a single stack or queue, respectively. Recently, the concepts of stack and queue layouts have been extended by using a double-ended queue or a restricted-input queue for storing the edges of a page. We extend this line of study to edge-weighted graphs by introducing priority queue layouts, that is, the edges on each page are stored in a priority queue whose keys are the edge weights. First, we show that there are edge-weighted graphs that require a linear number of priority queues. Second, we characterize the graphs that admit a priority queue layout with a single queue, regardless of the edge-weight function, and we provide an efficient recognition algorithm. Third, we show that the number of priority queues required independently of the edge-weight function is bounded by the pathwidth of the graph, but can be arbitrarily large already for graphs of treewidth two. Finally, we prove that determining the minimum number of priority queues is NP-complete if the linear ordering of the vertices is fixed.

Problem

• Extends stack and queue layouts to edge-weighted graphs using priority queues.
• Characterizes graphs requiring one priority queue regardless of edge weights.
• Shows NP-completeness of finding minimal priority queues for fixed vertex order.

Innovation

• Priority queue layouts for edge-weighted graphs
• Characterization of single-queue priority layouts
• NP-complete minimum priority queue determination