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arXiv · 2311.06563

Color Structures and the Monotone Satisfiability Problem with Bounded Variable Occurrence

Hannah Van Santvliet·Ronald de Haan
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Abstract

We study \textsc{Monotone 3-Sat-$(\leq k,1)$}, a restricted variant of the \textsc{Satisfiability} problem where clauses consist of three variables and are monotone (every clause contains either only unnegated or only negated variables) with up to $k$ positive and exactly one negative occurrence per variable in the formula. We resolve a challenge posed by Darmann and Döcker (On simplified NP-complete variants of \textsc{Monotone} 3-\textsc{Sat}, Discrete Applied Mathematics 292:45--58, 2021) by proving that for~$k\in \{3,4\}$, the problem is trivial in the sense that every instance satisfying the given restrictions is satisfiable. This result closes the remaining gap in a dichotomy theorem: Triviality for $k\in \{1,2\}$ follows by a result by Tovey (A simplified NP-complete satisfiability problem, Discrete Applied Mathematics 8(1):85--89, 1984), while NP-completeness for~$k\geq 5$ was shown by Darmann and Döcker. To obtain our result, we introduce the notion of \emph{color structures} and show that a satisfying assignment can always be constructed in $\mathcal{O}(n \cdot m)$ time, where $n$ and $m$ denote the number of negative and positive clauses of the input formula, respectively.

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Hannah Van Santvliet, Ronald de Haan. 2026-07-15. Color Structures and the Monotone Satisfiability Problem with Bounded Variable Occurrence. https://arxiv.org/abs/2311.06563

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