42
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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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31
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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22
votes
Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient
Computing a generating set of invariants (sometimes called the computational problem of "Noether's Normalization Lemma") for the action of $SL_3$ on an $n$-dimensional vector space $V$. (You ...
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17
votes
Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient
The task of unification went from an exponential solution to linear time in the timespan of about a decade. The original exponential algorithm was a corner-stone for symbolic AI approaches and enabled ...
- 271
16
votes
Accepted
NP-hard problems with very fast exponential-time algorithms
The desired property holds for Independent Set (and probably other problems) in graphs of suitably bounded tree width.
Fix any constant $\epsilon>0$ and consider the Independent Set problem ...
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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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12
votes
Accepted
EXP-Complete Problems vs Subexponential Algorithms
Due to popular demand, I’m converting my comment to an answer.
A simple padding argument shows that for every constant $\epsilon>0$, there exist EXP-complete problems in $\mathrm{DTIME}(2^{n^\...
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12
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Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient
I can think of two additional examples to the ones mentioned above, although I'm not sure that they were ever considered intractable.
Lovász Local Lemma - The Lovász local lemma (LLL) is a powerful ...
12
votes
Accepted
What are some examples of decidable Natural Problems outside of EEXP?
Let me give a few examples in the form of decision procedures for natural first-order theories.
By a result of Berman [1], Presburger arithmetic $\mathrm{Th}(\mathbb N,+)$ (or $\mathrm{Th}(\mathbb Z,+,...
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10
votes
Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient
Interior point algorithms for LP. Although they came after Ellipsoid they are a different class of provably polynomial-time algorithms. And despite initial skepticism about their ability to outperform ...
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10
votes
Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient
Until Francis's QR algorithm was discovered, computing the eigenvalues was often done by first computing the characteristic polynomial, which was often an expensive and inaccurate endeavor, as has ...
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9
votes
Accepted
Does two-sided error have more capability than one-sided error?
This isn't my area, so many apologies if I say something incorrect:
1) "What evidence do we have that $\mathsf{BPP}\subseteq \mathsf{REXP}$?"
Isn't this unconditionally true? It should follow from $\...
- 995
8
votes
Accepted
What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?
I think for getting 1.99-approximation algorithm this paper by Manurangsi and Trevisan, has the current fastest algorithm.
- 397
8
votes
Accepted
Reference request: complexity of $k$-partite $k$-SAT
Claim: If there exists an $\epsilon > 0$ such that for every $k'$, $k'$-partite $k'$-SAT can be solved in $2^{n(1-\epsilon)}$ time, then SETH fails.
Proof:
Suppose such an algorithm exists. We ...
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7
votes
Oracle comparing $EXP$ with $UP$
The quoted Beigel, Buhrman, and Fortnow paper gives a solution to 2 in Theorem 1.8: there is an oracle relative to which $\mathrm{P=Mod_3P}$ (which implies $\mathrm{P=UP}$), and $\mathrm{\oplus P=NP=...
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7
votes
Accepted
Oracle comparing $EXP$ with $UP$
$\mathsf{UP} \neq \mathsf{EXP}$ is open. A UP-generic oracle* should make $\mathsf{P} \neq \mathsf{UP} = \mathsf{EXP}$, and since $\mathsf{UP} \subseteq \mathsf{\oplus P} \subseteq \mathsf{EXP}$ ...
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7
votes
Accepted
Consequences of faster parameterized integer programming
An instance of CNF-SAT with $k$ variables can easily be written as a 0/1 integer linear program over the same variable set, since a clause such as $x_1 \vee x_3 \vee \neg x_4 \vee \neg x_6$ naturally ...
- 5,285
6
votes
Accepted
Correctness of AKS algorithm for shortest vector problem
What you seem to be missing is that $\tau$ is not applied to all "green" vectors. Instead, think of every point $x_i$ as having a coin attached to it. Before you use $x_i$ in the algorithm, you toss ...
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6
votes
Accepted
The problem of deciding whether a monotone CNF implies a monotone DNF
Assuming SETH, the problem is not solvable in time $O\bigl(2^{(1-\epsilon)n}\mathrm{poly}(l)\bigr)$ for any $\epsilon>0$.
First, let me show that this is true for the more general problem where $\...
- 18.2k
6
votes
Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient
I wanted to say Linear Programming, but although theoretical algorithm that are as fast as matrix multiplication have now been found, in practice people are still mostly using the exponential Simplex ...
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6
votes
Accepted
Is there a W[1]-hard problem that can be solved in $2^{o(n)}$ time?
How about Planar Capacitated Dominating Set? It is W[1]-hard (see the paper by Bodlaender, Lokshtanov, Penninkx in IWPEC 2009), but should be solvable in $2^{O(\sqrt{n}\log n)}$ by using the fact that ...
- 3,722
6
votes
Most general setting for fine-grained exponential-time complexity classes?
(Just now noticed this question.) There are a lot of questions in the above question. I will try to just address the last few.
Might it be the case that a RAM program can solve general CNF-SAT in ...
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6
votes
Questions regarding SETH
Cygan, Kratsch and Nederlof give a $(2+\sqrt{2})^{\texttt{pw}} n^{O(1)}$ algorithm for hamiltonicity on graphs with $n$ vertices and pathwidth $\texttt{pw}$ (assuming you are given the pathwidth ...
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6
votes
Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient
Algorithms that computed the position of planets when they thought the earth was the center of the universe versus when they realized the sun was the center of the solar system. :-)
While that's not ...
6
votes
Accepted
Best known algorithm for NEXP-complete problem
For every $\epsilon>0$, there exists an NEXP-complete language $L_\epsilon$ in $\mathrm{NTIME}(2^{n^\epsilon})$, and therefore in $\mathrm{DTIME}(2^{2^{n^\epsilon}})$, which is below $2^{o(2^n)}$.
...
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5
votes
Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient
Perhaps a good example is Graph Isomorphism testing, also discussed here:
Fastest known deterministic algorithm for the undirected Graph Isomorphism problem
and here:
https://people.cs.uchicago.edu/~...
- 1,616
5
votes
What are some examples of decidable Natural Problems outside of EEXP?
The Petri nets/Vector Addition Systems reachability problem is not primitive recursive; more precisely, it is complete for the class $\mathcal F_\omega$ of the fast-growing hierarchy (essentially the ...
- 18.2k
4
votes
Problems with Unknown Single Exponential Time Agorithms
The problem of testing whether a given integer linear program $L$ with $n$ variables has a feasible solution can be solved using $n^{2.5n+o(n)}\cdot |L|$ arithmetic operations. It is a major open ...
- 1,934
4
votes
Problems with Unknown Single Exponential Time Agorithms
Tensor Isomorphism. The best-known algorithm for 3-Tensor Isomorphism over $\mathbb{F}_q$ takes time $q^{\Theta(n^2)}$, and over $\mathbb{R}$ or $\mathbb{C}$ takes times $2^{\Theta(n^2)}$. (The same ...
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4
votes
On $\text{ETH}$ with $m$ as parameter: consequences of algorithm running in time $2^{\delta m}$ where $\delta \to 0$ as $k \to \infty$
This is an answer to the updated question (the original question seems harder).
Let $\mu'_k$ be the smallest constant such that $k$-SAT that has clauses of length exactly $k$ and no trivial clauses ...
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