Elsevier

Discrete Applied Mathematics

Volume 292, 31 March 2021, Pages 45-58
Discrete Applied Mathematics

On simplified NP-complete variants of Monotone 3 -Sat

https://doi.org/10.1016/j.dam.2020.12.010Get rights and content

Abstract

We consider simplified versions of 3 -Sat, the variant of the famous Satisfiability Problem where each clause is made up of exactly three distinct literals formed over pairwise distinct variables. More precisely, the focus of this work is laid on Monotone 3 -Sat, the restriction of 3 -Sat to formulas with monotone clauses, where a clause is monotone if it contains only unnegated variables or only negated variables. We prove several hardness results for Monotone 3 -Sat with respect to a variety of restrictions imposed on the variable appearances.

In particular, we show that for any k2, Monotone 3 -Sat turns out to be NP-complete even if each variable appears exactly k times unnegated and exactly k times negated. Therewith, for Monotone 3 -Sat with balanced variable appearances we establish a sharp boundary between NP-complete and polynomial time solvable cases.

In addition, we prove that for any k5, Monotone 3 -Sat is NP-complete even if each variable appears exactly k times unnegated and exactly once negated. 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. (2018) 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. (2003).

Introduction

The famous Boolean satisfiability problem, and in particular 3 -Satisfiability, can be considered the classical decision problem in computer science. 3 -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 3 -Satisfiability.

In 3 -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 3 -Sat as the variant of 3 -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 3 -Sat, the restriction of 3 -Sat to formulas in which each clause is monotone, i.e., contains only unnegated or only negated variables. It is known that Monotone 3 -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 p times unnegated and q times negated, for every fixed pair (p,q) with p2 and q2,

  • (1b)

    each variable appears exactly k times unnegated and once negated, for every fixed integer k5, 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 k times unnegated and k times negated, for each k2. 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 3 -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) 3 -Satisfiability. Both Planar 3 -Satisfiability and Planar Monotone 3 -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 3 -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 3 -Sat without affecting satisfiability. In Section 4 we provide hardness results for Monotone 3 -Sat in the restricted setting of balanced variable appearances, where each variable appears unnegated and negated equally often. In fact, we show that Monotone 3 -Sat is NP-complete if each variable appears exactly p times unnegated and q times negated, for every fixed pair (p,q) with p2 and q2. In Section 5 Monotone 3 -Sat is analyzed restricted to instances in which each variable appears exactly once negated. Then we consider Monotone 3 -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 V={x1,,xn} 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 {T,F}, where T represents true and F represents false. A literal is a variable or its negation, i.e., an element of LV={xi,xi¯xiV}. A clause is a subset of LV, and a k-clause contains exactly k 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 3 -Sat without affecting satisfiability, therefore, e.g., implying that Monotone 3 -Sat- (p,q) and Monotone 3 -Sat- (p,q) are equivalent in terms of NP-completeness.

Lemma 1

For every k1, 1, and r3, such

Monotone 3 -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 3 -Sat- (k,k) for k3. There, we begin with a simple corollary stating NP-completeness of Monotone 3 -Sat- (4,4), even in a restricted setting. Then we turn to Monotone 3 -Sat- (3,3) and, by the use of several lemmata, show its NP-completeness. Along the way, we in fact show that

Monotone 3 -Sat with exactly one negated appearance per variable

In this section Monotone 3 -Sat is analyzed restricted to instances in which each variable appears exactly once negated. We settle the computational complexity status of Monotone 3 -Sat- (k,1) for each fixed k5. We do not answer the question of its computational complexity for k{3,4}, which, to the best of our knowledge, is still open. However, for k{3,4} we can show that when restricted to a “small” number of unnegated appearances each instance of Monotone 3-Sat- (k,1) is satisfiable.

On a restricted variant of Monotone 3 -Sat-E4

Finally, we consider Monotone 3 -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 C(x,y) of monotone clauses, where Vaux={a,b,,h} are new variables.

  • 1.

    {ā,c̄,ē}

  • 2.

    {ā,c̄,f̄}

  • 3.

    {ā,d̄,ḡ}

  • 4.

    {b̄,c̄,h̄}

  • 5.

    {b̄,ē,ḡ}

  • 6.

    {b̄,f̄,ḡ}

  • 7.

    {d̄,ē,h̄}

  • 8.

    {d̄,f̄,h̄}

  • 9.

    {a,b,x}

  • 10.

    {c,d,x}

  • 11.

    {e,f,x}

  • 12.

    {g,h,y}

Then, a truth assignment β for {x,y} can be extended to a truth assignment β for {x,y}Vaux that satisfies C(x,y) if and only if β(v)=T for

Conclusion

We have shown that a restricted variant of Monotone 3 -Sat-E 4 is NP-complete, and that Monotone 3 -Sat- (2,2) is NP-complete. In addition, our results show that in fact Monotone 3 -Sat- (p,q) is NP-complete for each fixed pair (p,q){(r,s),(s,r)r2,s2}.

In particular, Monotone 3 -Sat- (k,k) is NP-complete for all k2. By a result of Tovey [21, Theorem 2.4] the latter problem is trivial for k=1, i.e., all such instances are satisfiable. Therewith, for Monotone 3 -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 3 -Sat- (2,2).

References (21)

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