
Monotone 3Sat(2,2) is NPcomplete
We show that Monotone 3Sat remains NPcomplete if (i) each clause conta...
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Positive Planar Satisfiability Problems under 3Connectivity Constraints
A 3SAT problem is called positive and planar if all the literals are po...
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Solving QSAT in sublinear depth
Among PSPACEcomplete problems, QSAT, or quantified SAT, is one of the m...
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On simplified NPcomplete variants of NotAllEqual 3Sat and 3Sat
We consider simplified, monotone versions of NotAllEqual 3Sat and 3S...
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Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3SAT
In directed graphs, we investigate the problems of finding: 1) a minimum...
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∀∃Rcompleteness and areauniversality
In the study of geometric problems, the complexity class ∃R turned out t...
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Expanding Visibility Polygons by Mirrors upto at least K units
We consider extending visibility polygon (VP) of a given point q (VP(q))...
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Placing quantified variants of 3SAT and NotAllEqual 3SAT in the polynomial hierarchy
The complexity of variants of 3SAT and NotAllEqual 3SAT is well studied. However, in contrast, very little is known about the complexity of the problems' quantified counterparts. In the first part of this paper, we show that ∀∃ 3SAT is Π_2^Pcomplete even if (1) each variable appears exactly twice unnegated and exactly twice negated, (2) each clause is a disjunction of exactly three distinct variables, and (3) the number of universal variables is equal to the number of existential variables. Furthermore, we show that the problem remains Π_2^Pcomplete if (1a) each universal variable appears exactly once unnegated and exactly once negated, (1b) each existential variable appears exactly twice unnegated and exactly twice negated, and (2) and (3) remain unchanged. On the other hand, the problem becomes NPcomplete for certain variants in which each universal variable appears exactly once. In the second part of the paper, we establish Π_2^Pcompleteness for ∀∃ NotAllEqual 3SAT even if (1') the Boolean formula is linear and monotone, (2') each universal variable appears exactly once and each existential variable appears exactly three times, and (3') each clause is a disjunction of exactly three distinct variables that contains at most one universal variable. On the positive side, we uncover variants of ∀∃ NotAllEqual 3SAT that are coNPcomplete or solvable in polynomial time.
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