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Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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0answers
440 views

Complexity Theory conferences? [closed]

What are the most significant annual Complexity Theory conferences? Rules: One conference per answer Include a link to the conference
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3answers
506 views

What evidence is there that $coRP \neq NP$?

What evidence is there that $coRP \neq NP$? $coRP$ is the class of languages for which there exists a probabililistic Turing Machine that runs in polynomial time and always answers Yes on an input ...
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3answers
508 views

$NP\cap coAM$ Languages

What other problems languages different than graph isomorphism are in $NP\cap coAM$? Can you give some references? Update: I forgot to mention that I'm interested in languages not known to be in $...
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4answers
1k views

“All-different hypergraph coloring” - known problem?

I am interested in the following problem: Given a set X and subsets X_1, ..., X_n of X, find a coloring of the elements of X with k colors such that the elements in each X_i are all differently ...
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4answers
360 views

Alternate notion of complexity based on gap between brute-force and the best algorithm?

Typically, efficient algorithms have a polynomial runtime and an exponentially-large solution space. This means that the problem must be easy in two senses: first, the problem can be solved in a ...
26
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3answers
909 views

Natural problems in $NP \cap coNP$ not in $UP \cap coUP$?

Are there any natural problems in $NP \cap coNP$ that are not (known to be/thought to be) in $UP \cap coUP$? Obviously the big one everyone knows about in $NP \cap coNP$ is the decision version of ...
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3answers
546 views

Two Variants of NP

Here are two variations on the definition of NP. They (almost certainly) define distinct complexity classes, but my question is: are there natural examples of problems that fit into these classes? (...
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5answers
1k views

Evidence that PPAD is hard?

There is often-quoted philosophical justification for believing that P != NP even without proof. Other complexity classes have evidence that they are distinct, because if not, there would be "...
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10answers
3k views

Kolmogorov complexity applications in computational complexity

Informally speaking, Kolmogorov complexity of a string $x$ is a length of a shortest program that outputs $x$. We can define a notion of 'random string' using it ($x$ is random if $K(x) \geq 0.99 |x|$)...
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2answers
740 views

Is APX contained in NP?

A problem P is said to be in APX if there exists some constant c > 0 such that a polynomial-time approximation algorithm exists for P with approximation factor 1 + c. APX contains PTAS (seen by ...
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9answers
2k views

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 ...
14
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3answers
535 views

How hard is it to reduce termination to partial correctness?

If you are familiar with program verification you are likely to prefer reading the Question before the Background. If you are not familiar with program verification then you may still be able to ...
20
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1answer
403 views

Explain Gurvits's tensor-rank interpretation of Deolalikar's paper

[Note: I believe this question in no way hinges on the correctness or incorrectness of Deolalikar's paper.] On Scott Aaronson's blog Shtetl Optimized, in the discussion about Deolalikar's recent ...
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2answers
584 views

Lower bounds on Gaussian complexity

Define the Gaussian complexity of an $n \times n$ matrix to be the minimal number of elementary row and column operations required to bring the matrix into upper-triangular form. This is a quantity ...
18
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1answer
560 views

Best known joint containments for/by NP and Parity-P?

Parity-P is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero ...
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4answers
2k views

Bounded-cardinality bounded-frequency set cover: hardness of approximation

Consider the minimum set cover problem with the following restrictions: each set contains at most $k$ elements and each element of the universe occurs in at most $f$ sets. Example: the case $k = 4$ ...
37
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3answers
1k views

Parameterized complexity of Hitting Set in finite VC-dimension

I'm interested in the parameterized complexity of what I'll call the d-Dimensional Hitting Set problem: given a range space (i.e. a set system / hypergraph) S = (X,R) having VC-dimension at most d and ...
27
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2answers
1k views

What are the consequences of Parity-L = P?

Parity-L is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero ...
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2answers
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 ...
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2answers
3k views
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3answers
713 views

Can it be determined if language L lies in NP?

Given a language L defined by a Turing Machine that decides it, is it possible to determine algorithmically whether L lies in NP?
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1answer
19k views

Is integer factorization an NP-complete problem? [duplicate]

Possible Duplicate: What are the consequences of factoring being NP-complete? What notable reference works have covered this?
38
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7answers
5k views

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?
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2answers
511 views

Lower bounds for constant-depth formulae?

We know a lot about the limitations of (polynomial size) constant-depth circuits. Since (polynomial size) constant-depth formulae are an even more restricted model of computation, all problems known ...
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3answers
488 views

Are recursive forms of Godel's statement possible?

The self-referentiality of the P/NP problem has sometimes been highlighted as a barrier to its resolution, see, for instance, Scott Aaronson's paper, is P vs. NP formally independent? One of the many ...
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5answers
1k views

Complexity of testing for a value versus computing a function

In general we know that the complexity of testing whether a function takes a particular value at a given input is easier than evaluating the function at that input. For example: Evaluating the ...
46
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4answers
3k views

What are the best current lower bounds on 3SAT?

What are the best current lower bounds for time and circuit depth for 3SAT?
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2answers
554 views

What evidence do we have for (and against) Unique Games Conjecture?

Subhash Khot's Unique Games Conjecture is one of active research areas in complexity theory. What evidence do we have for it? What evidence do we have against it?
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4answers
442 views

Which results in complexity theory make essential use of uniformity?

A complexity class separation proof uses uniformity of complexity classes essentially if the proof does not prove the result for nonuniform version, for example proofs based on diagonalization (like ...
125
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28answers
18k views

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? ...
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2answers
612 views

H-free cut problem

Suppose you are given a connected, simple, undirected graph H. The H-free cut problem is defined as follows: Given a simple, undirected graph G, is there a cut (partition of vertices into two ...
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4answers
1k views

What specific evidence is there for P = RP?

RP is the class of problems decidable by a nondeterministic Turing machine that terminates in polynomial time, but that is also allowed one-sided error. P is the usual class of problems decidable by ...
30
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2answers
2k views

Hierarchies in NP (under the assumption that P != NP)

Assuming that P != NP, I believe it has been shown that there are problems which are not in P and not NP-Complete. Graph Isomorphism is conjectured to be such a problem. Is there any evidence of more ...
23
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1answer
2k views

Consequences of Complete problems for NP intersects coNP

What are the consequences of having complete problems in $NP\cap coNP$?
64
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11answers
4k views

What are good references to understanding the proof of the PCP theorem?

I'm familiar with a lot of results that use the PCP theorem (mainly in approximating algorithms), but I've never come across a clear explanation of the PCP theorem (ie, that $\mathsf{NP} = \mathsf{PCP}...
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3answers
932 views

Graph Isomorphism and hidden subgroups

I'm trying to understand the relationship between graph isomorphism and the hidden subgroup problem. Is there a good reference for this ?
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4answers
842 views

Correspondence between complexity classes and logic

I took a class once on Computability and Logic. The material included a correlation between complexity / computability classes (R, RE, co-RE, P, NP, Logspace, ...) and Logics (Predicate calculus, ...
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3answers
3k views

Is Deolalikar's 2010 proof that $P \ne NP$ correct?

There was recently a claimed proof that $P \ne NP$. Not long after its publication there were raised some issues with this proof. So ... is the proof correct or not ? (Please only answer this if you ...
20
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3answers
462 views

space-bounded TMs and oracles

In general, the query-tape for an oracle counts towards the space-complexity of a TM. However, it seems plausible to allow a write-only oracle-tape (such as is used in L-space reductions). Is such a ...
14
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2answers
967 views

What is the following variation on Set Cover known as?

What is the following variation on set cover known as? Given a set S, a collection C of subsets of S and a positive integer K, do there exist K sets in C such that every pair of elements of S lies in ...