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Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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141 votes
30 answers
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Problems Between P and NPC

Factoring and graph isomorphism are problems in NP that are not known to be in P nor to be NP-Complete. What are some other (sufficiently different) natural problems that share this property? ...
76 votes
9 answers
11k views

Are runtime bounds in P decidable? (answer: no)

The question asked is whether the following question is decidable: Problem  Given an integer $k$ and Turing machine $M$ promised to be in P, is the runtime of $M$ ${O}(n^k)$ with respect to input ...
John Sidles's user avatar
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47 votes
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An NP-complete variant of factoring.

Arora and Barak's book presents factoring as the following problem: $\text{FACTORING} = \{\langle L, U, N \rangle \;|\; (\exists \text{ a prime } p \in \{L, \ldots, U\})[p | N]\}$ They add, further ...
Michaël Cadilhac's user avatar
37 votes
2 answers
6k views

Status of Impagliazzo's Worlds?

In 1995, Russell Impagliazzo proposed five complexity worlds: 1- Algorithmica: $P=NP$ with all the amazing consequences. 2- Heuristica: $NP$-complete problems are hard in the worst-case ($P \ne NP$) ...
Mohammad Al-Turkistany's user avatar
52 votes
20 answers
9k views

NP-hard problems on trees

Several optimization problems that are known to be NP-hard on general graphs are trivially solvable in polynomial time (some even in linear time) when the input graph is a tree. Examples include ...
Shiva Kintali's user avatar
93 votes
14 answers
22k views

What kind of mathematical background is needed for complexity theory?

I am currently an undergraduate student, bound to graduate this year. After graduation, I am considering to work towards a TCS master/PhD. I have begun wondering what fields of mathematics are ...
chazisop's user avatar
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51 votes
9 answers
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Best Upper Bounds on SAT

In another thread, Joe Fitzsimons asked about "the best current lower bounds on 3SAT." I'd like to go the other way: What's the best current upper bounds on 3SAT? In other words, what is the time ...
Sadeq Dousti's user avatar
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41 votes
2 answers
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Semantic vs. Syntactic Complexity Classes

In his "Computational Complexity" book, Papadimitriou writes: RP is in some sense a new and unusual kind of complexity class. Not any polynomially bounded nondeterministic Turing machine can be the ...
Sadeq Dousti's user avatar
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52 votes
4 answers
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What are the best current lower bounds on 3SAT?

What are the best current lower bounds for time and circuit depth for 3SAT?
Joe Fitzsimons's user avatar
48 votes
4 answers
5k views

Generalized Ladner's Theorem

Ladner's Theorem states that if P ≠ NP, then there is an infinite hierarchy of complexity classes strictly containing P and strictly contained in NP. The proof uses the completeness of SAT under many-...
András Salamon's user avatar
37 votes
5 answers
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NEXP-complete problems

There are tons of NP-complete problems around and sources collecting them, e.g. see the book by Garey and Johnson. I would be interested to see a list of NEXP-complete problems as well. Is there one ...
43 votes
2 answers
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Are the problems PRIMES, FACTORING known to be P-hard?

Let PRIMES (a.k.a. primality testing) be the problem: Given a natural number $n$, is $n$ a prime number? Let FACTORING be the problem: Given natural numbers $n$, $m$ with $1 \leq m \leq n$, ...
k m's user avatar
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23 votes
2 answers
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Problem in BPP but not known to be in RP or co-RP

Is there an example of a natural problem that's in BPP but that's not known to be in RP or co-RP?
arnab's user avatar
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43 votes
3 answers
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What are the reasons that researchers in computational geometry prefer the BSS/real-RAM model?

Background The computation over real numbers are more complicated than computation over natural numbers, since real numbers are infinite objects and there are uncountably many real numbers, therefore ...
Kaveh's user avatar
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43 votes
7 answers
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Many-one reductions vs. Turing reductions to define NPC

Why do most people prefer to use many-one reductions to define NP-completeness instead of, for instance, Turing reductions?
Matthias's user avatar
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33 votes
5 answers
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Programming languages for efficient computation

It is impossible to write a programming language that allows all machines that halt on all inputs and no others. However, it seems to be easy to define such a programming language for any standard ...
Artem Kaznatcheev's user avatar
28 votes
2 answers
1k views

Finding a prime greater than a given bound

Is a deterministic polynomial-time algorithm known for the following problem: Input: a natural number $n$ (in binary encoding) Output: a prime number $p > n$. (According to a list of open ...
Vincenzo's user avatar
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4 answers
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Proofs, Barriers and P vs NP

It is well known that any proof resolving the P vs NP question must overcome relativization, natural proofs and algebrization barriers. The following diagram partitions the "proof space" into ...
Shiva Kintali's user avatar
19 votes
1 answer
710 views

Looking for a nice problem inside SC but not in the first two levels

The complexity zoo doesn't have much about the $\mathsf{SC}$. I am looking for a nice$^\dagger$ problem that is in higher levels of the hierarchy, i.e. a problem in $\mathsf{DTimeSpace}(n^{O(1)},\lg^{...
Kaveh's user avatar
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65 votes
5 answers
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Problems that can be used to show polynomial-time hardness results

When designing an algorithm for a new problem, if I can't find a polynomial time algorithm after a while, I might try to prove it is NP-hard instead. If I succeed, I've explained why I couldn't find ...
Robin Kothari's user avatar
55 votes
2 answers
4k views

Can one amplify P=NP beyond P=PH?

In Descriptive Complexity, Immerman has Corollary 7.23. The following conditions are equivalent: 1. P = NP. 2. Over finite, ordered structures, FO(LFP) = SO. This can be thought of as "...
András Salamon's user avatar
40 votes
3 answers
4k views

Does $VP \neq VNP$ imply $P \neq NP$?

As far as I understand, the geometric complexity theory program attempts to separate $VP \neq VNP$ by proving that the permament of a complex-valued matrix is much harder to compute than the ...
Benno's user avatar
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37 votes
9 answers
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Surprising Results in Complexity (Not on the Complexity Blog List)

What were the most surprising results in complexity? I think it would be useful to have a list of unexpected/surprising results. This includes both results that were surprising and came out of ...
35 votes
2 answers
3k views

NTIME(n^k) ≠ DTIME(n^k) ?

In "On determinism versus nondeterminism and related problems" (Proc. IEEE FOCS, pages 429–438, 1983), Paul, Pippenger, Szemerédi and Trotter proved that $\mathsf{NTIME}(n)\neq\mathsf{DTIME}(n)$. ...
Bruno's user avatar
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32 votes
2 answers
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Derandomizing Valiant-Vazirani?

The Valiant-Vazirani theorem says that if there is a polynomial time algorithm (deterministic or randomized) for distinguishing between a SAT formula that has exactly one satisfying assignment, and an ...
Henry Yuen's user avatar
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31 votes
2 answers
7k views

What would be the consequences of factoring being NP-complete?

Are there any references covering this?
txwikinger's user avatar
31 votes
3 answers
2k views

Justification of log f in DTIME hierarchy theorem

If we look at DTIME hierarchy theorem, we've got a log due to the overhead in simulation of a deterministic Turing Machine by a universal machine : $DTIME(\frac{f}{\log f}) \subsetneq DTIME(f)$ We ...
Ludovic Patey's user avatar
29 votes
6 answers
2k views

Which SAT problems are easy?

What are "easy regions" for satisfiability? In other words, sufficient conditions for some SAT solver to be able to find a satisfying assignment, assuming it exists. One example is when each clause ...
Yaroslav Bulatov's user avatar
29 votes
7 answers
3k views

Proving lower bounds by proving upper bounds

The recent breakthrough circuit complexity lower-bound result of Ryan Williams provides a proof technique that uses upper-bound result to prove complexity lower-bounds. Suresh Venkat in his answer to ...
Mohammad Al-Turkistany's user avatar
22 votes
1 answer
878 views

How much computational power fits into a cubic centimeter?

This question is a followup on the question about DNA algorithms asked by Aadita Mehra. In comments there, Joe Fitzsimmons said, in part: [T]he radius of the system must scale proportionately to ...
Aaron Sterling's user avatar
20 votes
1 answer
1k views

Consequences of UP equals NP

EDIT at 2011/02/08: After some references finding and reading, I decided to separate the original question into two separate ones. Here's the part concerning UP vs NP, for the syntactic and semantic ...
Hsien-Chih Chang 張顯之's user avatar
20 votes
4 answers
3k views

Computational complexity in quantitative finance

Predicting the stock market is hard! Can TCS make this sentiment more formal? Recently I have started thinking a little bit about finance, and was wondering how knowledge of TCS could help. Hedge ...
Artem Kaznatcheev's user avatar
18 votes
2 answers
2k views

Is NP in $DTIME(n^{poly\log n})$?

Is NP in $DTIME(n^{poly\log n})$?
user avatar
17 votes
2 answers
744 views

Hardness of parameterized CLIQUE?

Let $0\le p\le 1$ and consider the decision problem CLIQUE$_p$ Input: integer $s$, graph $G$ with $t$ vertices and $\lceil p\binom{t}{2} \rceil$ edges Question: does $G$ contain a clique on at ...
András Salamon's user avatar
12 votes
2 answers
2k views

Is $coNP^{\#P}=NP^{\#P}=P^{\#P}$?

By http://www.cs.umd.edu/~jkatz/complexity/relativization.pdf If $A$ is a PSPACE-complete language, $P^{A}=NP^{A}$. If $B$ is a deterministic polynomial-time oracle, $P^{B}\ne NP^{B}$ (assuming $P\...
Mike Chen's user avatar
  • 709
9 votes
2 answers
2k views

(0,1)-vector XOR problem

this is a rewrite of another recent question of mine [1] that wasnt stated well (it had a semi obvious simplification, mea culpa) but I think theres still a nontrivial question at the heart of it. ...
vzn's user avatar
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50 votes
5 answers
7k views

Is the Chomsky-hierarchy outdated?

The Chomsky(–Schützenberger) hierarchy is used in textbooks of theoretical computer science, but it obviously only covers a very small fraction of formal languages (REG, CFL, CSL, RE) compared to the ...
Jakob's user avatar
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42 votes
4 answers
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Is $PH \subseteq PP$?

We know that the first level of the polynomial hierarchy (i.e. NP and co-NP) is in PP, and that $PP \subseteq PSPACE$. We also know from Toda's Theorem that $PH \subseteq P^{PP}$. Do we know whether $...
Huck Bennett's user avatar
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38 votes
3 answers
3k views

Complexity of exponential function

We know that the exponential function $\exp(x,y) = x^y$ over natural numbers is not computable in polynomial time, because the size of the output is not polynomially bounded in the size of the inputs. ...
Peter's user avatar
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33 votes
2 answers
2k views

When does "X is NP-complete" imply "#X is #P-complete"?

Let $X$ denote a (decision) problem in NP and let #$X$ denote its counting version. Under what conditions is it known that "X is NP-complete" $\implies$ "#X is #P-complete"? Of course the existence ...
Tyson Williams's user avatar
30 votes
3 answers
2k views

A decision problem which is not known to be in PH but will be in P if P=NP

Edit: As Ravi Boppana correctly pointed out in his answer and Scott Aaronson also added another example in his answer, the answer to this question turned out to be “yes” in a way which I had not ...
Tsuyoshi Ito's user avatar
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25 votes
2 answers
1k views

Is "Experimental Complexity Theory" being used to solve open problems?

Scott Aaronson proposed an interesting challange: can we use supercomputers today to help solve CS problems in the same way that physicists use large particle colliders? More concretely, my ...
Shane's user avatar
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23 votes
2 answers
1k views

How does the Mulmuley-Sohoni geometric approach to producing lower bounds avoid producing natural proofs (in the Razborov-Rudich sense)?

The exact phrasing of the title is due to Anand Kulkarni (who proposed this site be created). This question was asked as an example question, but I’m insanely curious. I know very little about ...
Ross Snider's user avatar
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23 votes
1 answer
1k views

Do all complexity classes have a leaf language characterization?

Leaf languages are a beautiful way to uniformly define many complexity classes. Most complexity classes are usually specified by a model of computation (e.g., deterministic/randomized TM), and a ...
Robin Kothari's user avatar
23 votes
2 answers
1k views

Relationship between symmetry and computational intractability?

The $k$-fixed point free automorphism problem asks for a graph automorphism which moves at least $k(n)$ nodes. The problem is $NP$-complete if $k(n)=n^c$ for any $c$>0. However, If $k(n)=O(\log n)$...
Mohammad Al-Turkistany's user avatar
21 votes
4 answers
2k views

DNA-algorithms and NP-completeness

What is the relationship between DNA-algorithms and the complexity classes defined using Turing machines? What do the complexity measures like time and space correspond to in DNA-algorithms? Can they ...
Aadita Mehra's user avatar
19 votes
5 answers
3k views

Why doesn't P=NP imply P=AP (i.e. P=PSPACE)?

It is well known that if $\mathbf{P}=\mathbf{NP}$ then the polynomial hierarchy collapses and $\mathbf{P}=\mathbf{PH}$. This can easily be understood inductively using oracle machines. The question ...
Joseph's user avatar
  • 387
19 votes
4 answers
3k views

What is the "right" definition of upper and lower bounds?

Let $f(n)$ be the worst case running time of a problem on input of size $n$. Let us make the problem a bit weird by fixing $f(n) = n^2$ for $n=2k$ but $f(n) = n$ for $n=2k+1$. So, what is the lower ...
Wei Yu's user avatar
  • 331
19 votes
3 answers
1k views

Trade off between time and query complexity

Working directly with time complexity or circuit lower bounds is scary. Hence, we develop tools like query complexity (or decision-tree complexity) to get a handle on lower bounds. Since each query ...
Artem Kaznatcheev's user avatar
16 votes
3 answers
5k views

Subset sum vs. Subset product (strong vs. weak NP hardness)

I was hoping that some one might be able to explain to me why exactly the subset product problem is strongly NP-hard while the subset sum problem is weakly NP-hard. Subset Sum: Given $X = \{x_1,...,...
RDN's user avatar
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