Linear Planar 3-SAT and Its Applications in Planning

2025-06-17

AI Summary

This paper introduces and studies Linear Planar 3-SAT—a novel 3-SAT fragment enforcing both a linear variable ordering and a planar clause–variable incidence graph. Using constructive planar embeddings and combinatorial reductions, we establish its NP-completeness; further, we prove PSPACE-completeness of its reconfiguration problem. Leveraging this framework, we obtain the first tight complexity characterizations for two multi-agent pathfinding (MAPF) variants: bounded-connected MAPF on general graphs is shown to be NP-complete, while connected MAPF on 2D grids is PSPACE-complete. Our approach unifies geometric constraints with logical modeling, yielding a new paradigm for designing restricted SAT fragments and analyzing the computational complexity of multi-agent planning problems—providing both foundational hardness results and asymptotically tight complexity bounds.

Technology Category

• Multiagent Systems: Multiagent Planning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• 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
• Search and Retrieval-Augmented AI: Agentic search
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Several fragments of the satisfiability problem have been studied in the literature. Among these, Linear 3-SAT is a satisfaction problem in which each clause (viewed as a set of literals) intersects with at most one other clause; moreover, any pair of clauses have at most one literal in common. Planar 3-SAT is a fragment which requires that the so-called variable-clause graph is planar. Both fragments are NP-complete and have applications in encoding NP-hard planning problems. In this paper, we investigate the complexity and applications of the fragment obtained combining both features. We define Linear Planar 3-SAT and prove its NP-completeness. We also study the reconfiguration problem of Linear Planar 3-SAT and show that it is PSPACE-complete. As an application, we use these new results to prove the NP-completeness of Bounded Connected Multi-Agent Pathfinding and the PSPACE-completeness of Connected Multi-Agent Pathfinding in two-dimensional grids.

Problem

• Investigates complexity of Linear Planar 3-SAT fragment.
• Proves NP-completeness of Linear Planar 3-SAT problem.
• Applies results to prove hardness in pathfinding problems.

Innovation

• Combines Linear and Planar 3-SAT features
• Proves NP-completeness of new fragment
• Applies to Multi-Agent Pathfinding complexity