40
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 ...
- 13.5k
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 ...
- 11.7k
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 ...
- 316
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 ...
- 21.5k
24
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 ...
- 504
22
votes
Accepted
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. ...
- 27.3k
22
votes
Accepted
Algebraic equivalent of SAT?
This is standard and widely used in computer science theory.
There are many references that use boolean polynomials with False -> 0 and True -> 1, or in other words, a polynomial over GF(2) used ...
- 11.7k
20
votes
Accepted
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. "...
- 2,164
19
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 ...
- 1,801
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 ...
- 803
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 ...
- 3,440
16
votes
Accepted
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 ...
- 2,164
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 ...
- 2,579
14
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 ...
- 490
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 ...
- 1,642
13
votes
Is this variation of TQBF still PSPACE-complete?
We proved that this game is PSPACE-complete for 5-CNFs but has Linear Time algorithm for 2-CNFs. The previous best result was Ahlroth and Orponen's 6-CNFs.
You can find the conference paper at ISAAC ...
13
votes
Accepted
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 ...
- 1,047
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 ...
- 16.9k
13
votes
Accepted
Complexity of 1-or-3-in-3-SAT (odd-3-SAT)
Somewhat surprisingly to me, this problem is in fact in PTIME.
The key insight is that, considering a clause $C$, letting $0 \leq k \leq 3$ be the number of negated literals in $C$, then the clause is ...
- 8,896
13
votes
Accepted
Is there a SETH (Strong Exponential Time Hypothesis) for CSP (Constraint Satisfaction Problem)?
This CSP is known to be SETH-hard. More precisely, assuming SETH, for any constant $\varepsilon > 0$ there is no $d^{(1-\varepsilon)n}$-time algorithm for solving this CSP with domain size $d$.
...
- 4,788
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 ...
- 50.9k
12
votes
Algebraic equivalent of SAT?
I think what you are asking about is also known as "polynomial calculus" in proof complexity and SAT solving. It was introduced in [1, 2] to investigate whether coNP can be separated from NP ...
- 11.4k
11
votes
On reducing the hardness of CNF-SAT to k-Clique
I don't know the answer to your specific question (it seems related to the question of whether W1=W[2]).
But the algorithm you give in your question is subsumed by several other results. Using your ...
- 27.3k
10
votes
Accepted
Random 3-SAT: What is the consensus experimental range of the threshold?
In light of the Ding--Sly--Sun verification of the 1-step Replica Symmetry Breaking picture for kSAT (when k is large enough) I think experts would now be pretty surprised if the MPZ/MMZ-conjectured ...
- 1,801
10
votes
Accepted
Improving Cook's generic reduction for Clique to SAT?
You can express $k$-clique as a SAT instance with $O(nk)$ variables and $O(nk^2)$ clauses. For fixed $k$, this is linear in $n$.
Let $x_{iv}=1$ if $v$ is the $i$th vertex in the clique (by ...
- 11.7k
10
votes
Accepted
Is bounded-width SAT decidable in logspace?
Indeed, using the resultss in Elberfeld-Jakoby-Tantau-2010 one can
show that SAT can be decided in logspace on formulas whose incidence graph has bounded treewidth. Here is a sketch of how the main ...
10
votes
Accepted
Sparsification Lemma for k-SAT and Exponential Time Hypothesis
I think your confusion might come from misquoting the statement of ETH.
$3$-SAT instances (with $n$ variables and $m$ clauses) cannot be solved in time $poly(n)\cdot 2^{o(n)}$.
This statement is ...
- 2,247
9
votes
Verifying a subtlety of Karp's original proof that SAT has a polynomial time reduction to 3SAT
The conjunction of the first two clauses, $(\sigma_1\cup\sigma_2\cup u_1)(\sigma_3\cup\ldots\cup\sigma_m\cup\bar{u}_1)$ is equisatisfiable to the original clause, as can be easily checked (any ...
- 2,500
9
votes
Complexity of 3SAT where each pair of 3-clauses share at most one variable
The central insight is that you can use two $2$-variable clauses to make sure two variables have the same value. Let me abbreviate $(x \vee \neg y) \wedge (y \vee \neg x)$ with $x=y$.
This allows the ...
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