DeepAI

# On simplified NP-complete variants of Not-All-Equal 3-Sat and 3-Sat

We consider simplified, monotone versions of Not-All-Equal 3-Sat and 3-Sat, variants of the famous Satisfiability Problem where each clause is made up of exactly three distinct literals. We show that Not-All-Equal 3-Sat remains NP-complete even if (1) each variable appears exactly four times, (2) there are no negations in the formula, and (3) the formula is linear, i.e., each pair of distinct clauses shares at most one variable. Concerning 3-Sat we prove several hardness results for monotone formulas with respect to a variety of restrictions imposed on the variable appearances. Monotone 3-Sat is the restriction of 3-Sat to monotone formulas, i.e. to formulas in which each clause contains only unnegated variables or only negated variables, respectively. In particular, we show that, for any k≥ 5, Monotone 3-Sat is NP-complete even if each variable appears exactly k times unnegated and exactly once negated. In addition, we show that Monotone 3-Sat is NP-complete even if each variable appears exactly three times unnegated and three times negated, respectively. In fact, we provide a complete analysis of Monotone 3-Sat with exactly six appearances per variable. Further, we prove that the problem remains NP-complete when restricted to instances in which each variable appears either exactly once unnegated and three times negated or the other way around. Thereby, we improve on a result by Darmann et al. [DDD18] showing NP-completeness for four appearances per variable. Our stronger result also implies that 3-Sat remains NP-complete even if each variable appears exactly three times unnegated and once negated, therewith complementing a result by Berman et al. [BKS03].

12/17/2019

### Monotone 3-Sat-(2,2) is NP-complete

We show that Monotone 3-Sat remains NP-complete if (i) each clause conta...
10/20/2017

### Planar 3-SAT with a Clause/Variable Cycle

In a Planar 3-SAT problem, we are given a 3-SAT formula together with it...
08/21/2019

### Existence and hardness of conveyor belts

An open problem of Manuel Abellanas asks whether every set of disjoint c...
09/06/2018

### Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT

In directed graphs, we investigate the problems of finding: 1) a minimum...
08/27/2021

### Positive Planar Satisfiability Problems under 3-Connectivity Constraints

A 3-SAT problem is called positive and planar if all the literals are po...
12/10/2021

### Beyond the Longest Letter-duplicated Subsequence Problem

Given a sequence S of length n, a letter-duplicated subsequence is a sub...
01/21/2022

### On the Satisfaction Probability of k-CNF Formulas

The satisfaction probability σ(ϕ) := _β:vars(ϕ) →{0,1}[βϕ] of a proposit...