[Paper Review] Monotone 3-Sat-4 is NP-complete

Andreas Darmann, Janosch Döcker|arXiv (Cornell University)|Mar 25, 2016

Formal Methods in Verification3 citations

TL;DR

This paper proves that Monotone 3-Sat-4, a restricted variant of the Boolean satisfiability problem where each clause contains exactly three distinct literals (all positive or all negative), and each variable appears in at most four clauses, is NP-complete. The proof constructs a satisfiability-preserving transformation using a 25-clause gadget that forces a designated variable to be true only when it is set to true, thereby eliminating 2-clauses while maintaining variable frequency bounds, thus resolving a long-standing conjecture about the complexity of this problem.

ABSTRACT

Monotone 3-Sat-4 is a variant of the satisfiability problem for boolean formulae in conjunctive normal form. In this variant, each clause contains exactly three literals---either all or none of them are positive, i.e., no clause contains both a positive and a negative literal---and every variable appears at most four times in the formula. Moreover, every clause consists of three distinct literals. We show that Monotone 3-Sat-4 is NP-complete.

Motivation & Objective

• To resolve a conjecture that Monotone 3-Sat-5 is NP-hard by proving a stronger result: Monotone 3-Sat-4 is NP-complete.
• To close the complexity gap between 3-Sat-3 (trivially satisfiable) and 3-Sat-5 (conjectured hard), showing that monotonicity does not reduce complexity below 4 occurrences per variable.
• To provide a satisfiability-preserving transformation that eliminates 2-clauses in monotone 3-Sat instances while ensuring no variable exceeds four appearances.
• To construct a minimal, equisatisfiable gadget with 25 clauses that enforces a variable to be true only when assigned true, using bounded variable frequency.

Proposed method

• Apply Gold’s replacement rule to convert mixed clauses in 3-Sat-4 into monotone (2,3)-Sat-4 instances, preserving equisatisfiability and variable frequency.
• Introduce a novel replacement rule, $\mathcal{R}_3$, to replace monotone 2-clauses with monotone 3-clauses using new variables that appear at most five times, avoiding variable frequency blowup.
• Design a 25-clause gadget $\mathcal{C}_z$ that is satisfiable if and only if a designated variable $z$ is set to true, with $z$ appearing exactly three times and all other variables at most four times.
• Use the gadget to replace negative 2-clauses $\{\bar{x}, \bar{y}\}$ by $\{\bar{x}, \bar{y}, \bar{z}\}$ and adding $\mathcal{C}_z$, ensuring equisatisfiability and bounded variable frequency.
• Prove the gadget’s correctness by contradiction: assume $z$ is false and derive a logical contradiction through clause dependencies, showing no satisfying assignment exists when $z$ is false.
• Verify that a valid truth assignment exists when $z$ is true (e.g., setting $\{z, g, a, r, e, p, k\}$ true and others false), confirming satisfiability.

Experimental results

Research questions

• RQ1 Is Monotone 3-Sat-4 NP-complete, despite the restriction that each variable appears in at most four clauses?
• RQ2 Can a satisfiability-preserving transformation eliminate 2-clauses in monotone 3-Sat instances while keeping variable frequency bounded at four?
• RQ3 Does a 25-clause gadget exist that enforces a variable to be true only when it is assigned true, with all variables appearing at most four times?
• RQ4 Is the conjecture that Monotone 3-Sat-5 is NP-hard valid, and can it be derived from the hardness of Monotone 3-Sat-4?
• RQ5 Can a monotone 3-Sat instance be transformed into an equisatisfiable 3-Sat instance with bounded variable frequency without increasing the number of clauses excessively?

Key findings

• Monotone 3-Sat-4 is proven to be NP-complete, resolving a conjecture attributed to Sarah Eisenstat from MIT’s Algorithmic Lower Bounds course.
• A 25-clause gadget $\mathcal{C}_z$ is constructed such that it is satisfiable if and only if the designated variable $z$ is set to true, and $z$ appears exactly three times.
• All variables in the gadget appear in at most four clauses, ensuring that variable frequency bounds are preserved during transformation.
• The gadget’s correctness is proven via contradiction: assuming $z$ is false leads to an unsatisfiable chain of implications across the clauses.
• A valid satisfying assignment exists when $z$ is set to true, confirming that the gadget is satisfiable under the intended condition.
• The result implies that Monotone 3-Sat-5 is also NP-complete, as Monotone 3-Sat-4 is a restriction of it, and the hardness of the former implies hardness of the latter.