Linear Planar 3-SAT and Its Applications in Planning

Victorien Desbois; Ocan Sankur; François Schwarzentruber

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arXiv · 2506.14713

Linear Planar 3-SAT and Its Applications in Planning

Victorien Desbois·Ocan Sankur·François Schwarzentruber

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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.

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Victorien Desbois, Ocan Sankur, François Schwarzentruber. 2025-06-17. Linear Planar 3-SAT and Its Applications in Planning. https://arxiv.org/abs/2506.14713

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