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Complexity status of monotone (3,1,4)-SAT

Determine the computational complexity of monotone (3,1,4)-SAT, specifically whether every monotone (3,1,4)-CNF is satisfiable or whether NP-hardness holds for this class.

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Background

Monotone SAT restricts every clause to contain either only positive or only negative literals. The paper shows NP-hardness for many (3,p,q)-SAT variants even under monotonicity, except for the case (3,1,3), which was recently proven to be universally satisfiable.

The status of monotone (3,1,4)-SAT remains unresolved: it is unknown whether all instances are satisfiable or whether the problem is NP-hard. Resolving this would complete the monotone analogue of the dichotomy for (3,1,q)-SAT.

References

The NP-hardness part of Theorem~\ref{the:dich} holds even for monotone SAT, where each clause is required to contain only positive or only negative literals, with the exception of monotone $(3,1,3)$-SAT, for which van Santvliet and de Haan have recently shown that all instances are satisfiable, and monotone $(3,1,4)$-SAT, which is still open.

— Small unsatisfiable $k$-CNFs with bounded literal occurrence (2405.16149 - Zhang et al., 25 May 2024) in Subsection “A dichotomy theorem for (3,1,q)-SAT”, after Theorem (Dichotomy)