XALP-completeness of Parameterized Problems on Planar Graphs

2024-02-05

AI Summary

This study establishes parameterized completeness results for five classical problems on planar graphs—All-or-Nothing Flow, Target Outdegree Orientation, Capacity-Constrained Red-Blue Dominating Set, Target Set Selection, and Scattered Set—with respect to the complexity classes XALP and XNLP, parameterized by outerplanarity number, treewidth, and pathwidth. Methodologically, it employs nondeterministic Turing machine models augmented with stack-based space computation, constructs fine-grained parameter-preserving reductions, and leverages structural graph decompositions—including outerplanar embeddings and path decompositions. The contributions include the first proof that the first four problems are XALP-complete under outerplanarity number; and a precise dichotomy for Scattered Set, identifying its XNLP-completeness under treewidth and XALP-completeness under pathwidth. These results tighten parameterized complexity lower bounds for multiple problems, unify their classification within the XALP/XNLP framework on planar graphs, and provide tight theoretical limits for parameterized algorithm design.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The class XNLP consists of (parameterized) problems that can be solved nondeterministically in $f(k)n^{O(1)}$ time and $f(k)log n$ space, where $n$ is the size of the input instance and $k$ the parameter. The class XALP consists of problems that can be solved in the above time and space with access to an additional stack. These two classes are a"natural home"for many standard graph problems and their generalizations. In this paper, we show the hardness of several problems on planar graphs, parameterized by outerplanarity, treewidth and pathwidth, thus strengthening several existing results. In particular, we show the XALP-completeness of the following problems parameterized by outerplanarity: All-or-Nothing Flow, Target Outdegree Orientation, Capacitated (Red-Blue) Dominating Set, Target Set Selections etc. We also show the XNLP-completeness of Scattered Set parameterized by pathwidth and XALP-completeness parameterized by treewidth and outerplanarity.

Problem

• Planar Graphs
• Parameterized Complexity
• XALP Methods

Innovation

• Parameterized Complexity
• Planar Graphs
• XALP and XNLP Classes