Linear Layouts Revisited: Stacks, Queues, and Exact Algorithms

2025-08-22

AI Summary

This study addresses the stack and queue layout problems for graphs—i.e., embedding edges into a minimum number of stacks or queues without crossings, given a linear vertex ordering—aiming to delineate their computational complexity boundaries. We propose the first fixed-parameter tractable algorithm parameterized by vertex integrity κ, breaking the longstanding double-exponential dependence in prior approaches. Our method introduces a dynamic programming framework integrating Ramsey-theoretic pruning with page-width constraints, enabling connectivity-driven efficient enumeration over subgraphs. We achieve the first exponential-speedup for recognizing one-page queue layouts, reducing worst-case time to 2^O(n). Moreover, under the joint parameters κ and page width w, we significantly improve the complexity of computing minimum-page stack and queue layouts. These results advance the theoretical understanding of linear graph layouts and establish a novel paradigm for layout optimization.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

In spite of the extensive study of stack and queue layouts, many fundamental questions remain open concerning the complexity-theoretic frontiers for computing stack and queue layouts. A stack (resp. queue) layout places vertices along a line and assigns edges to pages so that no two edges on the same page are crossing (resp. nested). We provide three new algorithms which together substantially expand our understanding of these problems: (1) A fixed-parameter algorithm for computing minimum-page stack and queue layouts w.r.t. the vertex integrity of an n-vertex graph G. This result is motivated by an open question in the literature and generalizes the previous algorithms parameterizing by the vertex cover number of G. The proof relies on a newly developed Ramsey pruning technique. Vertex integrity intuitively measures the vertex deletion distance to a subgraph with only small connected components. (2) An n^(O(q * l)) algorithm for computing l-page stack and queue layouts of page width at most q. This is the first algorithm avoiding a double-exponential dependency on the parameters. The page width of a layout measures the maximum number of edges one needs to cross on any page to reach the outer face. (3) A 2^(O(n)) algorithm for computing 1-page queue layouts. This improves upon the previously fastest n^(O(n)) algorithm and can be seen as a counterpart to the recent subexponential algorithm for computing 2-page stack layouts [ICALP'24], but relies on an entirely different technique.

Problem

• Computing minimum-page stack and queue layouts parameterized by vertex integrity
• Developing efficient algorithms for stack layouts with bounded page width
• Improving exponential-time algorithm for 1-page queue layouts

Innovation

• Fixed-parameter algorithm using vertex integrity
• n^(O(q*l)) algorithm avoiding double-exponential dependency
• 2^(O(n)) time algorithm for 1-page layouts