An even simpler hard variant of Not-All-Equal 3-SAT

TL;DR

This paper analyzes a monotone Not-All-Equal 3-SAT variant in which the clause set is the disjoint union of $k$ partitions of the variable set into 3-clauses and the overall collection is linear, denoting it Positive Linear $k$-Disjoint NAE-$3$-Sat-$E_k$. It proves a dichotomy: the problem is $NP$-complete for every fixed $k\ge 4$ but solvable in polynomial time for $k\le 3$ (with $k\in\{1,2\}$ yielding nae-satisfiable instances). The authors establish NP-hardness under both the non-monotone and linearity constraints by constructing EQ gadgets to enforce cross-partition consistency and by extending the reductions to larger $k$ via a $k$-to-$k+1$-partitions technique. They further translate these results to linear 3-uniform $k$-regular hypergraph bicolorability and discuss planarity implications, showing that planar instances become trivial under known constructions. The work also broadens the landscape with results on variable negation patterns, mixed clause sizes, and planarity, outlining a broader dichotomy and highlighting open questions for border cases (notably $k=3$).

Abstract

We show that Not-All-Equal 3-Sat remains NP-complete when restricted to instances that simultaneously satisfy the following properties: (i) The clauses are given as the disjoint union of k partitions, for any fixed $k \geq 4$, of the variable set into subsets of size 3, and (ii) each pair of distinct clauses shares at most one variable. Property (i) implies that each variable appears in exactly $k$ clauses and each clause consists of exactly 3 unnegated variables. Therewith, we improve upon our earlier result (Darmann and Döcker, 2020). Complementing the hardness result for at least $4$ partitions, we show that for $k\leq 3$ the corresponding decision problem is in P. In particular, for $k\in \{1,2\}$, all instances that satisfy Property (i) are nae-satisfiable. By the well-known correspondence between Not-All-Equal 3-Sat and hypergraph coloring, we obtain the following corollary of our results: For $k\geq 4$, Bicolorability is NP-complete for linear 3-uniform $k$-regular hypergraphs even if the edges are given as a decomposition into $k$ perfect matchings; with the same restrictions, for $k \leq 3$ Bicolorability is in P, and for $k \in \{1,2\}$ all such hypergraphs are bicolorable. Finally, we deduce from a construction in the work by Pilz (Pilz, 2019) that every instance of Positive Planar Not-All-Equal Sat with at least three distinct variables per clause is nae-satisfiable. Hence, when restricted to instances with a planar incidence graph, each of the above variants of Not-All-Equal 3-Sat turns into a trivial decision problem.

Theorems & Definitions (27)

• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Lemma 3.5
• proof