The Monotone Satisfiability Problem with Bounded Variable Appearances

Darmann, Andreas; Döcker, Janosch; Dorn, Britta

doi:10.1142/s0129054118500168

Cited by 9 publications

(7 citation statements)

References 13 publications

(14 reference statements)

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“…The same holds for instances of MONOTONE PLANAR 3-SAT with exactly three variables per clause. This solves an open problem by Darmann, Döcker, and Dorn [6,7], who show that the corresponding problem with at most three variables per clause remains NP-complete with bounds on the variable occurrences, which refines the result of de Berg and Khosravi [8].…”

Section: Resultssupporting

confidence: 71%

Planar 3-SAT with a Clause/Variable Cycle

Pilz

2017

Preprint

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In the PLANAR 3-SAT problem, we are given a 3-SAT formula together with its incidence graph, which is planar, and are asked whether this formula is satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it has been used as a starting point for a large number of reductions. In the course of this research, different restrictions on the incidence graph of the formula have been devised, for which the problem also remains hard.In this paper, we investigate the restriction in which we require that the incidence graph can be augmented by the edges of a Hamiltonian cycle that first passes through all variables and then through all clauses, in a way that the resulting graph is still planar. We show that the problem of deciding satisfiability of a 3-SAT formula remains NPcomplete even if the incidence graph is restricted in that way and the Hamiltonian cycle is given. This complements previous results demanding cycles only through either the variables or clauses.The problem remains hard for monotone formulas, as well as for instances with exactly three distinct variables per clause. In the course of this investigation, we show that monotone instances of PLANAR 3-SAT with exactly three distinct variables per clause are always satisfiable, thus settling the question by Darmann, Döcker, and Dorn on the complexity of this problem variant in a surprising way.

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Section: Resultssupporting

confidence: 71%

Planar 3-SAT with a Clause/Variable Cycle

Pilz

2017

Preprint

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“…We remark that the hardness result for condition (1d) improves upon the result for condition (1) by Darmann et al [DDD18,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).…”

Section: Introductionsupporting

confidence: 70%

“…Further related literature is concerned with the planar 3 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 [Lic82,Kra94] respectively [DBK12,DDD18]), while Pilz [Pil19,Theorem 11] shows that all instances of Planar Monotone 3-Sat, i.e., where each clause contains three distinct variables, are satisfiable. Moreover, the planar variant of Not-All-Equal 3-Sat can be solved in polynomial time [Mor88].…”

Section: Introductionmentioning

confidence: 99%

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

Darmann,

Döcker

2019

Preprint

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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].

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“…We will reduce from Monotone 3-Sat, a variant of 3-Sat, in which every clause contains only positive or only negative literals. It is known that this problem is NP-complete, even if each variable appears at most three times [11].…”

Section: Improving the Polynomial Resultsmentioning

confidence: 99%

Complexity of $C_k$-coloring in hereditary classes of graphs

Chudnovsky¹,

Huang²,

Rzążewski³

et al. 2020

Preprint

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For a graph F , a graph G is F -free if it does not contain an induced subgraph isomorphic to F . For two graphs G and H, an H-coloring of G is a mapping f : V (G) → V (H) such that for every edge uv ∈ E(G) it holds that f (u)f (v) ∈ E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F -free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P t -free graphs.We show that for every odd k ≥ 5 the C k -Coloring problem, even in the list variant, can be solved in polynomial time in P 9 -free graphs. The algorithm extends for the case of list version of C k -Coloring, where k is an even number of length at least 10.On the other hand, we prove that if some component of F is not a subgraph of a subdividecd claw, then the following problems are NP-complete in F -free graphs: a) extension version of C k -Coloring for every odd k ≥ 5, b) list version of C k -Coloring for every even k ≥ 6.

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The Monotone Satisfiability Problem with Bounded Variable Appearances

Cited by 9 publications

References 13 publications

Planar 3-SAT with a Clause/Variable Cycle

Planar 3-SAT with a Clause/Variable Cycle

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

Complexity of $C_k$-coloring in hereditary classes of graphs

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