Questions tagged [exp-time-algorithms]

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0answers
197 views

anything hinting that EXPTIME $\subseteqq$ PSPACE?

Anything or evidence hinting that $$EXPTIME \subseteqq PSPACE$$?
32
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12answers
5k views

Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient

This is somewhat of a meta-cstheory question, and is more historical in nature. What are some good examples of problems for which the literature followed the develpment below: The original algorithms,...
3
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0answers
132 views

If $\sf{E} = \sf{NE}$. Then $\sf{NP}-{P}$ contains no sparse sets [closed]

I am reading "The Complexity Companion" by Hemaspaandra & Ogihara, I have a question about lemma 1.21. In its proof, they suppose $L$ is some sparse language in $\sf{NP}$ ($||L^{=n}||&...
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1answer
313 views

On $\text{ETH}$ with $m$ as parameter: consequences of algorithm running in time $2^{\delta m}$ where $\delta \to 0$ as $k \to \infty$

It has been shown in [1] that $k\text{-SAT}$ has a $2^{o(n)}$ algorithm if and only if it has a $2^{o(m)}$ algorithm, $n$ being the number of variables and $m$ being the number of clauses. Being $s_k=\...
2
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0answers
157 views

Time complexity of Succinct-CVP

I want to know what is the best known lower time complexity of Succinct-CVP? The succinct version of many P-complete problems are EXP-complete and Succinct-CVP is EXP-complete too (It is because of ...
-1
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1answer
220 views

Does two-sided error have more capability than one-sided error?

From $P=RP$ extrapolation we might think $EXP=REXP$. What evidence do we have $BPP\subseteq REXP$? What consequence $REXP\subseteq BPP$ gives other than what $EXP\subseteq BPP$ gives?
15
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2answers
2k views

NP-hard problems with very fast exponential-time algorithms

NP-hard problems with very fast exact exponential-time algorithms, say with $O(1.01^n)$ time, are very rare. Is any fact like "For any constant $\epsilon>0$ there is an NP-hard 'natural' ...
6
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1answer
240 views

What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?

It is known that computing $(\sqrt 2 -\epsilon)$-approximation for VC is NP-hard and that UGC implies that even a $(2 -\epsilon)$-approximation is hard. There is also a parameterized algorithm for ...
5
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1answer
198 views

Correctness of AKS algorithm for shortest vector problem

Short question In the end of section 1 of Regev's notes about the AKS algorithm for SVP, why is the following true? for each such $i$,$y_i− x_i$ remains $w$ with probability $1/2$ or otherwise ...
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0answers
91 views

A question about UE

Much has been written about the class UP see related (even more in literature) example question here. Much is understood about the class UP, and its place in collapsing the PH too. UP has a played ...
1
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0answers
57 views

Asymptotic time required to simulate a Turing machine M for k steps

Problem: Given an encoding of a Turing machine M and a natural number k as input, find the output of M (given a blank tape) after k steps. Wikipedia's page on EXPTIME-complete says it takes O(k) time ...
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1answer
83 views

Question on deduction that a certain problem requires exponential space

My question concern's a statement from the classic paper The equivalence problem for regular expressions with squaring requires exponential space. Regular expressions with squaring are like ordinary ...
13
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1answer
412 views

The problem of deciding whether a monotone CNF implies a monotone DNF

Consider the following decision problem Input: A monotone CNF $\Phi$ and a monotone DNF $\Psi$. Question: Is $\Phi \to \Psi$ a tautology? Definitely you can solve this problem in $O(2^n \cdot \...
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1answer
380 views

Why is it a mystery if PSPACE ?= EXPTIME?

It seems obvious to me that $PSPACE \neq EXPTIME$. I, however, do not believe that my seemingly obvious logic would not be picked up by more intelligent people if it was so simple, so I'm assuming ...
7
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2answers
366 views

Most general setting for fine-grained exponential-time complexity classes?

Consider the class of functions computable in time $(b+o(1))^n = 2^{\log_2{(b)} \times n + o(n)}$ on a $2$-tape Turing machine. By the Hennie-Stearns theorem, the same functions are computable in ...
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1answer
160 views

Proof: PTIME not equal EXPTIME [closed]

Can someone give me the name of a paper where this is proved or maybe just prove it here (if it's easy enough)?
8
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0answers
107 views

Reference for a circuit lower bound for slightly superexponential time

It is known that $EXP$ doesn't have circuits of size $n^k$. On the other hand proving $10 n$ lower bound on circuit size for $E$, $NE$ or even $E^{NP}$ is a known open problem. My question is ...
5
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2answers
220 views

Oracle comparing $EXP$ with $UP$

Heller (Theorem 6) gave an oracle relative to which $NP=EXP$, and Homer & Selman gave an oracle relative to which $P=UP$ and $\Sigma_2^P=EXP$. Beigel, Buhrman, Fortnow (freely available author's ...
3
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1answer
141 views

Is it possible to approximate Maximum Independent Set in $O(2^k\text{poly}(n))$ time?

We know that MIS is hard to approximate within a $n^{1-\epsilon}$ factor in polynomial time and that it is $W[1]$-hard and thus unlikely to admit a $f(k)\text{poly}(n)$ time exact algorithm. (here, $k$...
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0answers
321 views

Intermediate problems between PSPACE and EXPTIME

Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves. Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, ...
5
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1answer
130 views

Reference request: complexity of $k$-partite $k$-SAT

Let's consider following variation of $k$-SAT that I will call $k$-partite $k$-SAT: given $n$ variables that are divided into $k$ groups and a $k$-SAT formula $\phi$ such that each clause has literal ...
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0answers
195 views

Parameterized Algorithm to Speed up Exact Exponential-time Algorithm

The connection between $c^kn^{O(1)}$ for $c<4$ and exact exponential-time algorithms beating brute-force $O(2^n)$ algorithms has been known for a long time. However, when $c\geq 4,$ there are not ...
3
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1answer
133 views

Consequences of faster parameterized integer programming

Integer programming in $k$ variables can be done in $k^{O(k)}$ time and $O(k^c)$ space. Is there any consequence if it can be done in $k^{O(k^\alpha)}$ time and $O(k^c)$ space for some $\alpha\in(0,1)...
4
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1answer
209 views

Is there a W[1]-hard problem that can be solved in $2^{o(n)}$ time?

This question is about subset problems (the solution is a subset of the instance, so trivially enumerable in $2^n \cdot n^c$ time), and the parameter is the solution size, so-called the standard ...
10
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1answer
288 views

EXP-Complete Problems vs Subexponential Algorithms

Does the fact that a problem $A$ is EXP-time complete implies that $A$ is not in $DTIME(2^{o(n)})$? I'm aware that by the time hierarchy theorem, $EXP=DTIME(2^{n^{O(1)}})$ is not included in $E=...
33
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2answers
4k views

Do any quantum algorithms improve on classical SAT?

Classical algorithms can solve 3-SAT in $1.3071^n$ time (randomized) or $1.3303^n$ time (deterministic). (Reference: Best Upper Bounds on SAT ) For comparison, using Grover's algorithm on a quantum ...
5
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2answers
329 views

Questions regarding SETH

I read about the strong exponential time hypothesis, which states (as far as I understand) that SAT problem cannot be solved in running time $O(2^{\epsilon n})$ for any $\epsilon < 1$, where $n$ is ...
14
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6answers
452 views

Problems with Unknown Single Exponential Time Agorithms

I'm looking for examples of problems for which it is easy to get algorithms running in time $2^{O(n\log n)}$, or $2^{O(n^c)}$ for some $c>1$ but for which it is not known whether there is an ...
8
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1answer
288 views

Two DFA intersection emptiness connections to SETH & L vs P

(re "fine grained complexity") Wehar has proved that Two DFA intersection emptiness in $O(n^{2-\epsilon})$ time → SETH false. does anyone see any particular key proof difficulty, challenge, ...
8
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1answer
427 views

Complexity of validity problem for Monadic First Order Logic?

Monadic First Order Logic is FOL with no function symbols, and predicate symbols restricted to arity 1. For this question, let's say that the = symbol is also forbidden. I want to know the complexity ...
4
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1answer
90 views

Approximate matching in table of integer vectors

Disclaimer: This is my first question on cstheory.stackexchange.com so please be forgiving. I have a list of M (M is big, more than 1 million elements) vectors of integers. Each vector can contain 0-...
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0answers
43 views

Information about algorithm to generate sequences

I want to make an application for generating a sequence (called S) of items (I), based on conditions (called C). The Conditions are defined as a property with a 'bonus/reduction'. The total score (T) ...
16
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0answers
435 views

What is the evidence for average case separation between EXP and NEXP?

There is significant evidence from cryptography that there exist NP-complete problems that are hard in the average case (meaning that e.g. $AvgP \nsupseteq DistNP$). Namely, we have candidate one-way ...
5
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0answers
243 views

Is MAX-SAT SETH (like) hard?

If we make a random assignment to the variables in $k$-sat ($m$ clauses), we are going to satisfy $(1-2^{-k})m$ clauses in expectation. In general satisfying fewer clauses is considered easy. There ...
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0answers
70 views

Finding exact value with a quotients of products of random values

Sorry for the haphazard title: really not sure what this should be called Suppose we have a set of $z$ random values $S = r_1, \dots, r_z$ drawn from $\mathbb{Z}_N$ (where $N$ is some large prime). ...
5
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1answer
2k views

What is the worst-case runtime complexity to transform a NFA to DFA via Rabin-Scott's power set construction?

What is the worst-case runtime complexity to transform a NFA to DFA via Rabin-Scott's power set construction? Why? Details: http://en.wikipedia.org/wiki/Powerset_construction states that the worst-...
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3answers
4k views

Examples of problems where exponential algorithms run faster than polynomial algorithms for practical sizes?

Do you know of any problems (preferably at least somewhat well known), where, for a practical problem size, an exponential algorithm runs much faster than a best-known polynomial time counterpart. ...
0
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1answer
212 views

Examples of $2^{\Theta(n^2)}\text{poly}(n)$-time algorithms

What are notable examples of problems for which the best currently known algorithm has $2^{\Theta(n^2)}\text{poly}(n)$ running time ?
4
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1answer
872 views

Example of context-free grammar that triggers exponential behaviour without memoization in RD parsers

It is often said that memoization brings the complexity of recursive-descent parsers from exponential to polynomial. However, I had a hard time finding an example grammar that triggers the exponential ...
12
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2answers
381 views

$\overline{SAT} \in NTIME(subexp)$?

Is it possible that $\overline{SAT} \in NTIME(\exp(n^{0.9}))$ ? Are there interesting consequences of such containment? Would it contradict the Exponential Time Hypothesis?
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0answers
904 views

What's the hardest problem with a non-trivial exact algorithm?

Exact algorithms for NP-complete problems are sometimes feasible, if the input is small enough. I’ve also came across some algorithms which are not practical even for very small inputs, and their ...
2
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1answer
268 views

Design of exact algorithms for non-local hard problems

In the connected dominating set problem (CDS) we are given an $n$-vertex undirected graph, and asked to find the smallest connected subset $S$ of vertices such that each vertex not in $S$ is adjacent ...
12
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1answer
516 views

Computational Model in SETH

Impagliazzo, Paturi and Calabro, Impagliazzo, Paturi introduced Exponential-Time Hypothesis (ETH) and Strongly Exponential-Time Hypothesis (SETH). Roughly, SETH says that there is no algorithm which ...
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2answers
473 views

Hardness of a subcase of Set Cover

How hard is the Set Cover problem if the number of elements is bounded by some function (e.g, $\log n$) where $n$ is the size of the problem instance. Formally, Let $\mathcal{U}=\{e_1, \cdots, e_m\}$...
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0answers
967 views

Exact algorithm for NAE-3SAT

The NAE-3SAT problem is to determine whether a given 3CNF formula has a satisfying assignment that gives each clause at least one false (and at least one true) literal. The problem is NP-complete. One ...
9
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2answers
394 views

Exact exponential-time algorithms for 0-1 programs with nonnegative data

Are there known algorithms for the following problem that beat the naive algorithm? Input: matrix $A$ and vectors $b,c$, where all entries of $A,b,c$ are nonnegative integers. Output: an optimal ...
14
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0answers
296 views

Exponential-time factorization of polynomials

Let an explicit field be a field for which equality is decidable (in some standard model of computation). I am interested in the factorization of univariate polynomials over an explicit field. It is ...
10
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2answers
600 views

Exact exponential-time algorithms for 0-1 programming

Are there known algorithms for the following problem that beat the naive algorithm? Input: A system $Ax \le b$ of $m$ linear inequalities. Output: A feasible solution $x^*\in \{0,1 \}^n$ if one ...
3
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0answers
247 views

Slightly Faster Exponential Algorithm for Integer Programming with Multi-linear Variables

Integer programing is one of the most narutal optimization tools. As an analogy of DNF or CNF in the Boolean function theory, we can consider the following equation. $x_{1}x_{2}x_{3}+$ $x_{3}x_{4}x_{5}...
13
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1answer
863 views

Exact algorithms for non-convex quadratic programming

This question is about quadratic programming problems with box constraints (box-QP), i.e., optimisation problems of the form minimise $f(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} + \mathbf{c}^T \mathbf{...