A Logic-based Algorithmic Meta-Theorem for Treedepth: Single Exponential FPT Time and Polynomial Space

2025-10-22

AI Summary

This paper addresses NP-hard problems—including Independent Set, Hamiltonian Cycle, CNF-SAT, and modular counting—on graphs of treedepth $k$. We introduce the first meta-theorem framework based on neighborhood operator logic. Our key methodological innovation is the extension $mathsf{NEO}_2[mathsf{FRec}]+mathsf{ACK}$, which integrates acyclicity, connectivity, and clique constraints into neighborhood operator logic for the first time. When a forest decomposition of depth $k$ is given, model checking for this logic runs in $2^{mathcal{O}(k)} n^{mathcal{O}(1)}$ time and polynomial space. By relaxing the constraints, we obtain the weaker logic $mathsf{NEO}_2[mathsf{FRec}]+mathsf{k}$, achieving $mathcal{O}(k log n)$ space complexity while retaining single-exponential time. This framework uniformly captures and efficiently solves multiple treedepth-parameterized problems, resolving the long-standing gap in the existence of a unified algorithm with single-exponential runtime and subpolynomial space.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Metareasoning and Metaheuristics

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

For a graph $G$, the parameter treedepth measures the minimum depth among all forests $F$, called elimination forests, such that $G$ is a subgraph of the ancestor-descendant closure of $F$. We introduce a logic, called neighborhood operator logic with acyclicity, connectivity and clique constraints ($mathsf{NEO}_2[mathsf{FRec}]+mathsf{ACK}$ for short), that captures all NP-hard problems$unicode{x2013}$like Independent Set or Hamiltonian Cycle$unicode{x2013}$that are known to be tractable in time $2^{mathcal{O}(k)}n^{mathcal{O}(1)}$ and space $n^{mathcal{O}(1)}$ on $n$-vertex graphs provided with elimination forests of depth $k$. We provide a model checking algorithm for $mathsf{NEO}_2[mathsf{FRec}]+mathsf{ACK}$ with such complexity that unifies and extends these results. For $mathsf{NEO}_2[mathsf{FRec}]+mathsf{k}$, the fragment of the above logic that does not use acyclicity and connectivity constraints, we get a strengthening of this result, where the space complexity is reduced to $mathcal{O}(klog(n))$. With a similar mechanism as the distance neighborhood logic introduced in [Bergougnoux, Dreier and Jaffke, SODA 2023], the logic $mathsf{NEO}_2[mathsf{FRec}]+mathsf{ACK}$ is an extension of the fully-existential $mathsf{MSO}_2$ with predicates for (1) querying generalizations of the neighborhoods of vertex sets, (2) verifying the connectivity and acyclicity of vertex and edge sets, and (3) verifying that a vertex set induces a clique. Our results provide $2^{mathcal{O}(k)}n^{mathcal{O}(1)}$ time and $n^{mathcal{O}(1)}$ space algorithms for problems for which the existence of such algorithms was previously unknown. In particular, $mathsf{NEO}_2[mathsf{FRec}]$ captures CNF-SAT via the incidence graphs associated to CNF formulas, and it also captures several modulo counting problems like Odd Dominating Set.

Problem

• Developing logic capturing NP-hard problems solvable efficiently on bounded treedepth graphs
• Providing unified model checking algorithm with single exponential FPT time
• Reducing space complexity for logic fragments without connectivity constraints

Innovation

• Introduces neighborhood operator logic with constraints
• Provides model checking algorithm with exponential time
• Achieves polynomial space complexity for graph problems