[Paper Review] On planar variants of the monotone satisfiability problem with bounded variable appearances

Andreas Darmann, Janosch Döcker+1 more|arXiv (Cornell University)|Apr 19, 2016

Computational Geometry and Mesh Generation2 references3 citations

TL;DR

This paper establishes the NP-completeness of several planar variants of the monotone satisfiability problem with bounded variable appearances, including cases where each variable appears exactly 3, 4, or 5 times and clauses are restricted to size 3. The authors use planar, orthogonal graph drawings and local gadget replacements to preserve planarity and variable degree bounds, enabling efficient reductions for NP-hardness proofs in computational complexity.

ABSTRACT

We show NP-completeness for several planar variants of the monotone satisfiability problem with bounded variable appearances. With one exception the presented variants have an associated bipartite graph where the vertex degree is bounded by at most four. Hence, a planar and orthogonal drawing for these graphs can be computed efficiently, which may turn out to be useful in reductions using these variants as a starting point for proving some decision problem to be NP-hard.

Motivation & Objective

• To establish NP-completeness for planar monotone satisfiability variants under bounded variable appearances.
• To ensure that the associated bipartite graphs have bounded vertex degrees (≤4) and remain planar.
• To develop efficient reductions using planar and orthogonal graph drawings for NP-hardness proofs.
• To explore whether monotone 3-SAT with bounded variable appearances remains NP-hard when each variable appears exactly s times, for s ≥ 4.
• To construct local replacement gadgets that preserve planarity and variable degree constraints during reductions.

Proposed method

• Reduction from Lichtenstein’s Planar 3-Sat to prove NP-hardness of Planar Monotone (2,3)-Sat-3.
• Application of Gold’s rule to transform mixed clauses into monotone clauses while preserving planarity and variable appearance bounds.
• Use of local gadget replacements for variable vertices in planar graphs to adjust variable appearances without breaking planarity.
• Construction of orthogonal drawings for the resulting graphs using known linear-time algorithms.
• Duplication of literals within clauses to ensure each clause contains exactly three literals and each variable appears exactly s times.
• Design of degree-3 and degree-4 variable gadgets that maintain biconnected and planar graph structures.

Experimental results

Research questions

• RQ1 Is Planar Monotone 3-Sat-4 NP-complete when each variable appears at most four times?
• RQ2 Can the monotone satisfiability problem remain NP-complete under exact variable appearance constraints (e.g., exactly 3, 4, or 5 times) in planar instances?
• RQ3 Does there exist a minimal s ≥ 4 such that Planar Monotone 3-Sat-s is NP-complete?
• RQ4 Can planar and orthogonal graph drawings be efficiently computed for monotone satisfiability instances with bounded variable appearances?
• RQ5 Can local gadget replacements preserve planarity and variable degree bounds during reductions?

Key findings

• Planar Monotone (2,3)-Sat-3 is NP-complete, even when each variable appears at most three times.
• Planar Monotone (2,3)-Sat-E3 is NP-complete, where each variable appears exactly three times.
• Planar Monotone 3-Sat*-E5 is NP-complete, with each variable appearing exactly five times and the graph being biconnected.
• The variant with exactly four variable appearances per variable and exactly three distinct literals per clause is NP-complete.
• All constructed instances have associated bipartite graphs with maximum vertex degree four, enabling efficient planar and orthogonal graph drawings.
• The reduction framework supports efficient construction of equisatisfiable instances with controlled variable appearances and planar structure.