# Tagged Questions

SAT stands for the Boolean satisfiability problem.

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### Given oracle for Max-3SAT compute clauses that cannot be satisfied

We know that Max-3SAT is NP-hard to compute exactly (and also hard to approximate to a particular constant multiplicative factor). However, suppose you are given an oracle for Max-3SAT that tells you ...
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### Asymptotic complexity of CDCL SAT solver that only selects negative literals

If a CDCL SAT solver only selects negative literals as decision literals (but can set positive literals through unit propogation) but has a perfect heuristic for determining which literal to select ...
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### Complexity of 3SAT where each pair of 3-clauses share at most one variable

Consider the variant of the 3SAT problem with the following restrictions: Each clause has 2 or 3 literals. Each pair of 3-literal clauses have at most one common variable. What is the complexity ...
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### Complexity of #SAT for monotone DNF formulae whose hypergraph is a hypertree

A monotone DNF on variables $x_1, \ldots, x_n$ is a disjunction of clauses, each clause being a conjunction of some of the $x_1, \ldots, x_n$. The #SAT problem asks, given a monotone DNF $\Phi$, how ...
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### On reducing the hardness of CNF-SAT to k-Clique

CNF-SAT refers to the following problem: Given a boolean formula $\phi$ in conjunctive normal form, does there exist an assignment to the variables that satisfies $\phi$. There are several ...
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### Verifying a subtlety of Karp's original proof that SAT has a polynomial time reduction to 3SAT

Stated briefly, my question is: is Karp's original proof reducing SAT to 3SAT unnecessarily elaborate? The details are as follows. In his 1972 paper Reducibility Among Combinatorial Problems, Karp ...
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### Balanced Max-2-SAT NP-Hardness

The Balanced Max-2-SAT is a special case of Max-2-SAT (each clause is a disjunction of exactly 2 literals) in which for every variable $x$, there is a $k$ such that $x$ appears positive exactly $k$ ...
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### “k-Swap SAT” problem

I would like to know if the following NP-complete problem has a name and has been studied: Input: Given a CNF formula $\varphi$ on $n$ variables, a truth assignment $\sigma:[n] \to \{T,F\}$ and an ...
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### Generate connected induced subgraphs as the satisfying assignments to a SAT instance

I want a SAT instance (in CNF) whose set of satisfying assignments are the connected induced subgraphs of a given input graph. A general solution would be helpful, but I really only need this when the ...
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### Improving Cook's generic reduction for Clique to SAT?

I am interested in reducing $k$-Clique to SAT without making the instance much larger. Clique is in NP so it can be reduced to SAT using logarithmic space. The straightforward Garey/Johnson textbook ...
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### Can we confirm that 2-SAT can indeed be transformed into Horn-SAT in this manner?

In the question, Translating SAT to HornSAT, Martin Seymour gives a method due to Joshua Grochow. It transforms 2-SAT into Horn-SAT, by creating a variable for every possible 2-SAT clause. Then, if ...
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### How is the MA version of SETH proven to be false?

According to this paper, which discusses a nondeterministic extension of the Strong Exponential Time Hypothesis (SETH), "[…] Williams has recently shown related hypotheses about Merlin-Arthur ...
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### #2-SAT or 3-SAT and variable that appears most often

Has anyone explored running times of 3-SAT or #2-SAT given by the occurrences of the highest occurring variable? In other words, if the variable that appears most often appears $x$ times, has anyone ...
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### Does solving matrix multiplication in quadratic time imply that SETH is false?

I have a little conjecture that if you could perform matrix multiplication (or solve 3-clique) in $O(n^2 \log(n))$ time, then you could solve CNF-SAT in $O(2^{(1-\epsilon)n})$ time. In other words, ...
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### Abstract high-level framwork for #SAT

In Abstract DPLL and some other sources there is a high-level framework/ model explained using states and transitions. I need (to build) such a model for a #SAT algorithm. I do know that #SAT ...
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### Is there a non-deterministic linear time algorithm for CNF-SAT?

The decision problem CNF-SAT can be described as follows: Input: A boolean formula $\phi$ in conjunctive normal form. Question: Does there exist a variable assignment that satisfies $\phi$? I'm ...
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### Complexity of QBF with Restrictions on Models [closed]

Do you know the complexity of the following decision problem? Given a quantified boolean formula (QBF) $\phi$ with $2n$ free variables with $n\in\mathbb{N}$. Is there a satisfying assignment s.t. ...
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### Best SAT upper bounds based on number of clauses

What's the best upper bounds based on number of clauses? In this question shown fastest algorithms for SAT, but there bounds depends from number of variables ( $O(const^n)$ where n is number of ...
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### pseudo boolean modularity constraint?

I have a constraint like d >= a ^ b ^ c, where a,b,c,d are binary, ^ is XOR. Is this a pseudo boolean modularity constraint or not? Most Pseudo boolean modularity constraints I saw are with equality, ...
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### Space requirements for solving True Quantified Boolean Formulas problem [closed]

I came across this section on the wikipedia page for the TQBF solving problem, and just can't wrap my head about the fact that the space requirement is linear. Moreover, it does not provide any ...
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### What is the simplest known solver for a np-complete problem?

Lets define the simpler of two terms as the one with shortest description length on the untyped λ-calculus. Trying to find the simplest solver for a np-complete problem, I've got this: ...
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### Complexity of generating a pseudo-Boolean function

A pseudo-Boolean function is a mapping from $\mathcal{B}^n = \{0, 1\}^n$ to $\mathbb{R}$. Following is a pseudo-Boolean function. s_1 s_4 - s_2 s_3 - s_3 s_5 - s_2 s_5 + s_1 + s_4 - s_1 s_3 - ...
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### Limited number of variable occurrences in 1-in-3 SAT

Is there a known result on complexity class of 1-in-3-SAT with restricted number of variable occurrences? I've come up with the following parsimonious reduction with Peter Nightingale, but I want ...
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### Converting Partial Weighted Max SAT to CIRCUIT SAT

I am interested in converting Partial Weighted Max SAT to SAT. I have been recommended to go through CIRCUIT SAT. Partial Weighted Max SAT consists of a set of hard clauses and a set of weighted ...
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### Does 1-in-3 SAT remain NP-hard even if every variable occurs both positively and negatively?

The standard problem 1-in-3 SAT (or XSAT or X3SAT) is: Instance: a CNF formula with every clause containing exactly 3 literals Question: is there a satisfying assignment setting precisely 1 literal ...
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### Is the computation of a satisfying variable assignment for a Boolean formula $FP^{NP}$-hard?

By the well-known self-reducibility of SAT we can obtain a satisfying variable assignment for a Boolean formula by a polynomial number of calls to an $NP$ oracle (delivering only yes/no answers). Thus,...
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### What's the upper bounds for #3-SAT circuits?

We have, from this thread on 3-SAT upper bounds, and this answer on #P that the current best upper bounds for 3-SAT is faster than $O(1.31)^n$, and approximately $O(1.64^n)$ for #3-SAT. Can we do ...
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### Properties expressible in 2-CNF or 2-SAT

How does one show that a certain property cannot be expressed in 2-CNF (2-SAT)? Are there any games, such as pebble games? It seems that the classical black pebble game and the black-white pebble ...
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### How many sets of vectors can be represented as the solutions of a Horn-SAT instance?

Let the solution space of a SAT instance be the set of Boolean vectors of satisfying assignments of $\{0,1\}$ to the variables (that result in the formula evaluating to TRUE). In other words, a ...
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### Best current space lower bound for SAT?

Following on from a previous question, what are the best current space lower bounds for SAT? With a space lower bound I here mean the number of worktape cells used by a Turing machine which uses ...
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### Conversion between k-SAT and XOR-SAT

According to XOR Satisfiability Solver Module for DPLL Integration by Tero Laitinen, we need $2^{n-1}$ CNF clauses to convert an $n$ literal XOR-SAT clause if we do not want to increase the number of ...
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### Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$

I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER ...
Deciding if a quantified boolean formula such as $\forall x_1 \exists x_2 \forall x_3\cdots \exists x_n \varphi(x_1, x_2,\ldots , x_n),$ always evaluates to true is a classical PSPACE-complete ...