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52 votes

Constraint satisfaction problem (CSP) vs. satisfiability modulo theory (SMT); with a coda on constraint programming

SAT, CP, SMT, (much of) ASP all deal with the same set of combinatorial optimisation problems. However, they come at these problems from different angles and with different toolboxes. These ...
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38 votes

Do any quantum algorithms improve on classical SAT?

Indeed, as wwjohnsmith1 said, you can get a square root speed-up over Schöning's algorithm for 3-SAT, but also more generally for Schöning's algorithm for k-SAT. In fact, many randomized algorithms ...
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37 votes
Accepted

Is this variation of TQBF still PSPACE-complete?

It is an Unordered Constraint Satisfaction game and it is PSPACE-complete and it has been proved to be PSPACE-complete only recently; a proof can be found in: Lauri Ahlroth and Pekka Orponen, ...
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31 votes

Are there any hard instances of 3-SAT when the clauses can only use literals that are "nearby" each other?

No. If the 3-SAT instance has $m$ clauses, then you can test satisfiability in $O(m 2^N)$ time. Since $N$ is a fixed constant, this is a polynomial-time algorithm that solves all instances of your ...
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  • 10.4k
30 votes

Is this variation of TQBF still PSPACE-complete?

It may also be worthwhile to note that this problem was also solved in the 70's by Thomas Schaefer in Complexity of decision problems based on finite two-person perfect-information games. In fact, ...
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30 votes
Accepted

Do any quantum algorithms improve on classical SAT?

I think one can obtain a non-trivial upper bound from quantum computing by speeding up the randomized algorithms of Schöning for 3-SAT. The algorithm of Schöning runs in time $(4/3)^n$ and ...
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27 votes

Theoretical explanations for practical success of SAT solvers?

I am assuming that you are referring to CDCL SAT solvers on benchmark data sets like those used in the SAT Competition. These programs are based on many heuristics and lots of optimization. There were ...
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  • 21.3k
26 votes
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Is there an oracle such that SAT is not infinitely often in sub-exponential time?

You can just take the oracle A s.t. NP$^A$=EXP$^A$ since EXP is not in i.o.-subexp. For SAT$^A$ it depends on the encoding, for example if the only valid SAT instances have even length then it is easy ...
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22 votes
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How is the MA version of SETH proven to be false?

You can find a preprint by following this link http://eccc.hpi-web.de/report/2016/002/ EDIT (1/24) By request, here is a quick summary, taken from the paper itself, but glossing over many things. ...
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22 votes

Theoretical explanations for practical success of SAT solvers?

I am typing this quite quickly due to severe time constraints (and didn't even get to responding earlier for the same reason), but I thought I would try to at least chip in with my two cents. I ...
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20 votes
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Counting the number of satisfying assignments in a POSITIVE CNF-SAT

This is still #P-complete [1]. This problem is usually referred to as montone (#)SAT. Monotone #2-SAT is already #P-complete (this is equivalent to counting vertex covers of a graph). [1] Roth, Dan. "...
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  • 1,855
19 votes
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Properties expressible in 2-CNF or 2-SAT

A family of bitvectors is the class of solutions to a 2-SAT problem if and only if it has the median property: if you apply the bitwise majority function to any three solutions you get another ...
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18 votes

Theoretical explanations for practical success of SAT solvers?

I'm not an expert in this area, but I think the random SAT / phase transition stuff is more or less completely unrelated to the industrial/practical applications stuff. E.g., the very good solvers ...
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18 votes

Theoretical explanations for practical success of SAT solvers?

Let me add my two cents of understanding to this, even though I've never actually worked in the area. You're asking one of two questions, "what are all the known approaches to proving theoretical ...
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17 votes
Accepted

MAX 1 in 2 SAT Algorithm

A monotone 1-in-2 clause demands that the two variables have different values. Thus, you can model the problem as a graph problem, with one vertex per variable which is to be colored black or white, ...
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17 votes

Theoretical explanations for practical success of SAT solvers?

There is a paper "Relating Proof Complexity Measures and Practical Hardness of SAT" by Matti Järvisalo, Arie Matsliah, Jakob Nordström, and Stanislav Živný in CP '12 that attempts to link the hardness ...
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17 votes

Are there any hard instances of 3-SAT when the clauses can only use literals that are "nearby" each other?

Incident graph of a SAT formula is a bipartite graph that has a vertex for each clause and each variable. We add edges between a clause and all of its variables. If the incident graph has bounded ...
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  • 3,420
16 votes

Is there an oracle such that SAT is not infinitely often in sub-exponential time?

You don't have to go to the lengths Lance was suggesting. For example, relative to a random oracle, using the oracle as a one-way function (say, evaluated on consecutive bit postions) is ...
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16 votes
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Complexity of the search version of 2-SAT assuming $\mathsf{L = NL}$

Given a satisfiable 2-CNF $\phi$, you can compute a particular satisfying assignment $e$ by an NL-function (that is, there is an NL-predicate $P(\phi,i)$ that tells you whether $e(x_i)$ is true). One ...
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15 votes

Counting the number of satisfying assignments in a POSITIVE CNF-SAT

This problem is Monotone-SAT. It is #P-Complete under Cook Reductions. It is one of those problems that are "easy to decide but hard to count." I recommend the following paper. Self-Reducibility of ...
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  • 2,579
14 votes

MAX 1 in 2 SAT Algorithm

An exact algorithm for the Max Monotone 1 in 2 Sat problem (i.e., MaxCut) running faster than $2^n$ (about $O(1.8^n)$ time) can be found in Chapter 6 of my PhD thesis, here: http://web.stanford.edu/~...
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14 votes
Accepted

Are there subexponential algorithms for PLANAR SAT known?

Well, you can apply the planar separator theorem together with dynamic programming and get running time $2^{O(\sqrt{n})}$, where $n$ is the number of vertices in the graph. The idea being that you try ...
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14 votes

Questions regarding SETH

The subtlety comes in where we introduce the notion of "harder". The reduction showing that SAT can be reduced to Hamiltonian Cycle shows that the latter is "harder" up to polynomial factors. In doing ...
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  • 1,624
14 votes
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On theoretical aproaches for solving $\mathsf{SAT}$ in special cases

Concerning Question 1, there have mainly been two lines of work to find tractable restrictions of SAT. The first one that you are already familiar with is to restrict the types of the clauses that ...
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  • 1,855
13 votes
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Translation of context-free parsing into SAT

(I guess the important word in the original question is ``published''.) There is such an encoding of context-free parsing (more exactly of CYK-style parsing) in Roland Axelsson, Keijo Heljanko, and ...
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  • 3,324
13 votes
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What are the consequences of solving XOR 3-SAT in Logspace?

Take a look at https://www.sciencedirect.com/science/article/pii/S0022000008001141 "The complexity of satisfiability problems: Refining Schaefer's theorem" by Allender et al. which answer your ...
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  • 1,027
13 votes
Accepted

What is the complexity of checking equivalence of two boolean formulae without NOT symbol?

Note that formulas using $\land$ and $\lor$ gates (and possibly the constants $0$ and $1$) are known as monotone. The complexity of monotone formula equivalence depends on how complex formulas are ...
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12 votes
Accepted

Best Upper Bounds on SAT

The best algorithm for 3-SAT now has numerical upper bound $O^{*}(1.306995^n)$ on unique-3-SAT and on general-3-SAT it is also fastest but now the specific values have not been analyzed yet. Authors ...
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  • 470
12 votes
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Approximating #P-hard problems

We're interested in additive approximations to #3SAT. i.e. given a 3CNF $\phi$ on $n$ variables count the number of satisfying assignments (call this $a$) up to additive error $k$. Here are some ...
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  • 2,743
12 votes

Definition of Planar 3-SAT

There's a nice compilation of definitions of related NP-complete planar satisfiability problems at http://courses.csail.mit.edu/6.890/fall14/scribe/lec7.pdf One of them, planar monotone 3-sat, allows ...
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