On simplified NP-complete variants of Monotone -Sat
Introduction
The famous Boolean satisfiability problem, and in particular -Satisfiability, can be considered the classical decision problem in computer science. -Satisfiability has been the first problem shown to be NP-complete decades ago (Cook [3]) and is of undisputed theoretical and practical importance1 ; it both appears in practical applications of routing, scheduling and artificial intelligence (see, e.g., Devlin and O’Sullivan [7], Nam et al. [17], Horbach et al. [12], and Kautz and Selman [13]), and is the most prominent problem, and probably the most frequently used one, for complexity analysis of decision problems. Therefore, it has continuously attracted researchers through decades focusing on the computational complexity of variants of the satisfiability problem (for recent work see, e.g., Pilz [19] or Paulusma and Szeider [18]).
In this paper, we add to that branch of literature and investigate the computational complexity2 of restricted variants of -Satisfiability.
In -Satisfiability, we are given a set of propositional variables and a collection of clauses, where each clause contains three literals. The question is whether there is a satisfying truth assignment, i.e., whether we can satisfy all clauses by assigning truth values to the variables.
In what follows, we will refer to -Sat as the variant of -Satisfiability in which each clause contains exactly three distinct literals formed over pairwise distinct variables—which is the setting we focus on in this paper.
The focus of this paper is laid on Monotone -Sat, the restriction of -Sat to formulas in which each clause is monotone, i.e., contains only unnegated or only negated variables. It is known that Monotone -Sat is NP-complete [11], [15], and that intractability holds even if (1) each variable appears exactly four times [6, Corollary 4]. We show that this problem remains NP-complete even if condition (1) is replaced by either one of the following three conditions:
- (1a)
each variable appears exactly times unnegated and times negated, for every fixed pair with and ,
- (1b)
each variable appears exactly times unnegated and once negated, for every fixed integer , or
- (1c)
each variable appears exactly three times unnegated and once negated or three times negated and once unnegated.
We point out that (1a) includes the case of balanced variable appearances, i.e, the case of each variable appearing exactly times unnegated and times negated, for each . Further, we remark that the hardness results for conditions (1a) and (1c) improve upon the result for condition (1) by Darmann et al. [6, Corollary 4]. Also, as a by-product, we derive the result that the classical -Sat problem remains NP-complete even if each variable appears exactly three times unnegated and once negated (observe that this implies hardness also for the vice versa case where each variable appears exactly once unnegated and three times negated). Therewith, we complement results of Tovey [21] and Berman et al. [2]: The former showed that 3-Sat remains NP-complete even if each variable appears in at most four clauses and it is trivial if the number of variable appearances is bounded by 3 [21, Theorem 2.3 and Theorem 2.4]; Berman et al. [2, Theorem 1] added to that result by showing that NP-completeness holds even if each variable appears exactly twice unnegated and twice negated.
Further related literature is concerned with the planar3 variants of (Monotone) -Satisfiability. Both Planar -Satisfiability and Planar Monotone -Satisfiability are known to be NP-complete even in restricted settings (e.g., see [14], [16] respectively [1], [6]), while Pilz [19, Theorem 11] shows that all instances of Planar Monotone -Sat, i.e., where each clause contains three distinct literals formed over pairwise distinct variables, are satisfiable.
The paper is structured as follows. In Section 2 we introduce basic notation and formally state the considered decision problems. In Section 3 we present a tool for increasing the number of literal appearances in an instance of Monotone -Sat without affecting satisfiability. In Section 4 we provide hardness results for Monotone -Sat in the restricted setting of balanced variable appearances, where each variable appears unnegated and negated equally often. In fact, we show that Monotone -Sat is NP-complete if each variable appears exactly times unnegated and times negated, for every fixed pair with and . In Section 5 Monotone -Sat is analyzed restricted to instances in which each variable appears exactly once negated. Then we consider Monotone -Sat restricted to instances in which each variable appears either three times unnegated and once negated or once unnegated and three times negated in Section 6. Finally, Section 7 concludes the paper with a concise summary of the results and a challenge for future research.
Preliminary versions of the results in this paper are contained in the online preprints [4], [8] on arXiv.
Section snippets
Preliminaries
Let be a set of propositional variables. For the remainder of the paper we simply say variable instead of propositional variable since all variables take on values in , where represents true and represents false. A literal is a variable or its negation, i.e., an element of . A clause is a subset of , and a -clause contains exactly distinct literals. Further, a clause is monotone if either all contained variables are negated (negative clause) or none of
Increasing the number of literal appearances in Monotone 3-SAT
We thank an anonymous referee for providing the following two lemmata (along with the below proof and a description) which are very helpful for improving the presentation of the paper. The lemmata establish a tool for increasing the number of literal appearances in any instance of Monotone -Sat without affecting satisfiability, therefore, e.g., implying that Monotone -Sat- and Monotone -Sat- are equivalent in terms of NP-completeness.
Lemma 1 For every , , and , such
Monotone -Sat with balanced variable appearances
In this section we consider the case of balanced variable appearances, where each variable appears unnegated and negated equally often; the section is structured as follows.
Section 4.1 is dedicated to Monotone -Sat- for . There, we begin with a simple corollary stating NP-completeness of Monotone -Sat- , even in a restricted setting. Then we turn to Monotone -Sat- and, by the use of several lemmata, show its NP-completeness. Along the way, we in fact show that
Monotone -Sat with exactly one negated appearance per variable
In this section Monotone -Sat is analyzed restricted to instances in which each variable appears exactly once negated. We settle the computational complexity status of Monotone -Sat- for each fixed . We do not answer the question of its computational complexity for , which, to the best of our knowledge, is still open. However, for we can show that when restricted to a “small” number of unnegated appearances each instance of Monotone 3-Sat- is satisfiable.
On a restricted variant of Monotone -Sat-E4
Finally, we consider Monotone -Sat-E4, i.e., with exactly four appearances of each variable. We begin this short section with the following lemma.
Lemma 6 Consider the following collection of monotone clauses, where are new variables.
Then, a truth assignment for can be extended to a truth assignment for that satisfies if and only if for
Conclusion
We have shown that a restricted variant of Monotone -Sat-E is NP-complete, and that Monotone -Sat- is NP-complete. In addition, our results show that in fact Monotone -Sat- is NP-complete for each fixed pair .
In particular, Monotone -Sat- is NP-complete for all . By a result of Tovey [21, Theorem 2.4] the latter problem is trivial for , i.e., all such instances are satisfiable. Therewith, for Monotone -Sat with
CRediT authorship contribution statement
Andreas Darmann: Conceptualization, Writing - review & editing, Writing - original draft. Janosch Döcker: Conceptualization, Writing - review & editing, Writing - original draft.
Acknowledgments
The authors are grateful for the diligent report and the useful hints – in particular for the findings presented in Section 3 – of an anonymous referee which improved (and shortened) the presentation of the paper significantly. In addition, the authors would like to thank Britta Dorn for insightful comments on a draft containing the construction of an unsatisfiable instance of Monotone -Sat- .
References (21)
- et al.
On a simple hard variant of Not-All-Equal 3-SAT
Theoret. Comput. Sci.
(2020) A special planar satisfiability problem and a consequence of its NP-completeness
Discrete Appl. Math.
(1994)- et al.
On the parameterized complexity of (k,s)-SAT
Inform. Process. Lett.
(2019) A simplified NP-complete satisfiability problem
Discrete Appl. Math.
(1984)- et al.
Optimal binary space partitions for segments in the plane
Internat. J. Comput. Geom. Appl.
(2012) - et al.
Approximation hardness of short symmetric instances of MAX-3SAT
Electronic Colloquium on Computational ComplexityReport No. 49
(2003) The complexity of theorem-proving procedures
- et al.
On simplified NP-complete variants of Not-All-Equal 3-SAT and 3-SAT
(2019) - et al.
The monotone satisfiability problem with bounded variable appearances
Internat. J. Found Comput. Sci.
(2018) - D. Devlin, B. O’Sullivan, Satisfiability as a classification problem, in: Proceedings of the 19th Irish Conference on...
Cited by (15)
Weighted Random k Satisfiability for k=1,2 (r2SAT) in Discrete Hopfield Neural Network
2022, Applied Soft ComputingPolarised random κ-SAT
2023, Combinatorics Probability and ComputingLocally Rainbow Paths
2024, arXivMinimizing Corners in Colored Rectilinear Grids
2024, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)