Questions tagged [complexity-classes]

Computational complexity classes and their relations

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Is Norbert Blum's 2017 proof that $P \ne NP$ correct?

Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...
Warren Schudy's user avatar
63 votes
8 answers
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What is the complexity class most closely associated with what the human mind can accomplish quickly?

This question is something I've wondered about for a while. When people describe the P vs. NP problem, they often compare the class NP to creativity. They note that composing a Mozart-quality ...
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55 votes
2 answers
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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
49 votes
4 answers
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What are the consequences of $\mathsf{L}^2 \subseteq \mathsf{P}$?

We know that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{P}$ and that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{L}^2 \subseteq $ $\mathsf{polyL}$, where $\mathsf{L}^2 = \mathsf{...
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44 votes
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P and NP classes explanation through lambda-calculus

In the introduction and explanation P and NP complexity classes often given through Turing machine. One of the model of computation is the lambda-calculus. I understand, that all of models of ...
Simplex'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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39 votes
7 answers
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What do we know about provably correct programs?

The ever increasing complexity of computer programs and the increasingly crucial position computers have in our society leaves me wondering why we still don't collectively use programming languages in ...
Alex ten Brink's user avatar
39 votes
3 answers
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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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4 answers
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Examples where the uniqueness of the solution makes it easier to find

The complexity class $\mathsf{UP}$ consists of those $\mathsf{NP}$-problems that can be decided by a polynomial time nondeterministic Turing machine which has at most one accepting computational path. ...
Andras Farago's user avatar
37 votes
8 answers
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Problems with big open complexity gaps

This question is about problems for which there is a big open complexity gap between known lower bound and upper bound, but not because of open problems on complexity classes themselves. To be more ...
Denis's user avatar
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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 ...
36 votes
2 answers
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Consequences of $SAT \in BQP$

As a TCS amateur, I'm reading some popular, very introductory material on quantum computing. Here are the few elementary bits of information I've learned so far: Quantum computers are not known to ...
Giorgio Camerani's user avatar
35 votes
3 answers
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complexity of greatest common divisor (gcd)

Consider the following counting problem (or the associated decision problem): Given two positive integers encoded in binary, compute their greatest common divisor (gcd). What is the smallest ...
Felix Breuer's user avatar
35 votes
1 answer
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$BQP$ vs $QMA$?

The central problem of complexity theory is arguably $P$ vs $NP$. However, since Nature is quantum, it would seem more natural to consider the classes $BQP$ (ie decision problems solvable by a ...
Anthony Leverrier's user avatar
35 votes
3 answers
1k views

An Anthology of Complexity Assumptions

In the paper The Random Oracle Hypothesis Is False, the authors (Chang, Chor, Goldreich, Hartmanis, Håstad, Ranjan, and Rohatgi) discuss the implications of the random-oracle hypothesis. They argue ...
34 votes
5 answers
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The unreasonable power of non-uniformity

From the common sense point of view, it is easy to believe that adding non-determinism to $\mathsf{P}$ significantly extends its power, i.e., $\mathsf{NP}$ is much larger than $\mathsf{P}$. After all,...
Andras Farago's user avatar
34 votes
5 answers
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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 "...
Aaron Roth's user avatar
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31 votes
4 answers
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Consequences of NP=PSPACE

What would be the nasty consequences of NP=PSPACE? I am surprised I did not found anything on this, given that these classes are among the most famous ones. In particular, would it have any ...
Denis's user avatar
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30 votes
0 answers
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The complexity of checking whether two DAG have the same number of topological sorts

This problem is highly related to the CNF one. Here is the problem: given two DAG (directed acyclic graphs), if they have the same counting of topological sorts, answer "Yes", otherwise, answer "No". ...
Mike Chen's user avatar
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29 votes
2 answers
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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 ...
Niel de Beaudrap's user avatar
29 votes
4 answers
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If P = NP were true, would quantum computers be useful?

Suppose that P = NP is true. Would there then be any practical application to building a quantum computer such as solving certain problems faster, or would any such improvement be irrelevant based on ...
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29 votes
2 answers
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Ladner's Theorem vs. Schaefer's Theorem

While reading the article "Is it Time to Declare Victory in Counting Complexity?" over at the "Godel's Lost Letter and P=NP" blog, they mentioned the dichotomy for CSP's. After some link following, ...
user834's user avatar
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Is NPI contained in P/poly?

It is conjectured that $\mathsf{NP} \nsubseteq \mathsf{P}/\text{poly}$ since the converse would imply $\mathsf{PH} = \Sigma_2$. Ladner's theorem establishes that if $\mathsf{P} \ne \mathsf{NP}$ then $\...
Vanessa's user avatar
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Does $EXP\neq ZPP$ imply sub-exponential simulation of BPP or NP?

By simulation I mean in the Impaglazzio-Widgerson [IW98] sense, i.e. sub-exponential deterministic simulation which appears correct i.o to every efficient adversary. I think this is a proof: if $EXP\...
Sebastian Ben Daniel's user avatar
28 votes
3 answers
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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 ...
Joshua Grochow's user avatar
28 votes
1 answer
1k views

Is uniform RNC contained in polylog space?

Log-space-uniform NC is contained in deterministic polylog space (sometimes written PolyL). Is log-space-uniform RNC also in this class? The standard randomized version of PolyL should be in PolyL, ...
Ryan O'Donnell's user avatar
27 votes
3 answers
1k views

Succinct Problems in $\mathsf{P}$

The study of Succinct representation of graphs was initiated by Galperin and Wigderson in a paper from 1983, where they prove that for many simple problems like finding a triangle in a graph, the ...
Nikhil's user avatar
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26 votes
3 answers
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Optimization problems with minimax characterization, but no polynomial-time algorithm

Consider optimization problems of the following form. Let $f(x)$ be a polynomial-time computable function that maps a string $x$ into a rational number. The optimization problem is this: what is the ...
Andras Farago's user avatar
26 votes
3 answers
2k views

Consequences of #P = FP

Which would be the consequences of #P = FP? I'm interested in both practical and theoretical consequences. From a practical point of view, I'm particularly interested in consequences on Artificial ...
Giorgio Camerani's user avatar
26 votes
2 answers
975 views

What are the consequences of $L = \oplus L$?

Shiva Kintali has just announced a (cool!) result that graph isomorphism for bounded treewidth graphs of width $\geq 4$ is $\oplus L$-hard. Informally, my question is, "How hard is that?" We know ...
Aaron Sterling's user avatar
25 votes
7 answers
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Is there a natural problem in quasi-polynomial time, but not in polynomial time?

László Babai recently proved that the Graph Isomorphism problem is in quasipolynomial time. See also his talk at University of Chicago, note from the talks by Jeremy Kun GLL post 1, GLL post 2, GLL ...
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25 votes
4 answers
3k views

Why do equalities between complexity classes translate upwards and not downwards?

Hey Guys, I understand that the padding trick allows us to translate complexity classes upwards - for example $P=NP \rightarrow EXP=NEXP$. Padding works by "inflating" the input, running the ...
gabgoh's user avatar
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25 votes
3 answers
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Constructivity in Natural Proof and Geometric Complexity

Recently, Ryan Willams proved that Constructivity in Natural Proof is unavoidable to derive a separation of complexity classes : $\mathsf{NEXP}$ and $\mathsf{TC}^{0}$. Constructivity in Natural ...
auyun's user avatar
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25 votes
3 answers
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Hardness of approximation - additive error

There is a rich literature and at least one very good book setting out the known hardness of approximation results for NP-hard problems in the context of multiplicative error (e.g. 2-approximation for ...
Simd's user avatar
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25 votes
1 answer
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What is $\mathsf{NP}$ restricted to linear size witnesses?

This is related to the question Is the Witness Size of Membership for Every NP Language Already Known? Some natural $\mathsf{NP}$(-complete) problems have linear length witnesses: a satisfying ...
argentpepper's user avatar
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25 votes
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What are the relationships between those hypotheses in Fine-Grained Complexity Theory?

Complexity theory, through such concepts as NP-completeness, distinguishes between computational problems that have relatively efficient solutions and those that are intractable. "Fine-grained" ...
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25 votes
1 answer
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Tardos Function Counterexample to Blum's $P\neq NP$ Claim

In this thread, Norbet Blum's attempted $P \neq NP$ proof is succinctly disproved by noting that the Tardos function is a counterexample to Theorem 6. Theorem 6: Let $f \in \mathcal{B}_n$ be any ...
user144527's user avatar
24 votes
3 answers
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Is BQP equal to BPP with access to an Abelian hidden subgroup oracle?

Is BQP equal to BPP with access to an Abelian hidden subgroup oracle?
Jason's user avatar
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2 answers
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What is the big version of NC?

$\mathsf{NC}$ captures the idea of efficiently parallelizable, and one interpretation of it is problems that are solvable in time $O(\log^c n)$ using $O(n^k)$ parallel processors for some constants $c$...
Artem Kaznatcheev's user avatar
24 votes
6 answers
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Statements that imply $\mathbf{P}\neq \mathbf{NP}$

This is sort of an open-ended question - for which I apologize in advance. Are there examples of statements that (seemingly) don't have anything to do with complexity or Turing machines but the ...
Dominic van der Zypen's user avatar
24 votes
1 answer
934 views

Is P equal to the intersection of all superpolynomial time classes?

Let us call a function $f(n)$ superpolynomial if $\lim_{n\rightarrow\infty} n^c/f(n)=0$ holds for every $c>0$. It is clear that for any language $L\in {\mathsf P}$ it holds that $L\in {\mathsf {...
Andras Farago's user avatar
23 votes
4 answers
2k views

Communication Complexity ...Classes?

Discussion: I've been spending some personal time lately learning various things in communication complexity. For instance, I've re-familiarized myself with the relevant chapter in Arora/Barak, ...
Daniel Apon's user avatar
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4 answers
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Can typed lambda calculi express *all* algorithms below a given complexity?

I know that the complexity of most varieties of typed lambda calculi without the Y combinator primitive is bounded, i.e. only functions of bounded complexity can be expressed, with the bound becoming ...
jkff's user avatar
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23 votes
4 answers
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What are the compelling reasons for believing $L\neq P$?

What are the compelling reasons for believing $L\neq P$? L is the class of log-space algorithms with pointers to the input. Suppose L=P for the moment. What would a log-space algorithm for a P-...
Jumer's user avatar
  • 239
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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23 votes
1 answer
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Is there any justification to believe that $NL\neq L$?

I wonder if there is any justification to believe that $NL=L$ or to believe that $NL\neq L$? It is known that $NL \subset L^2$. The literature on derandomization of $RL$ is pretty convincing that $RL=...
Klim's user avatar
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23 votes
1 answer
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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
22 votes
3 answers
1k views

Problems outside of P that are not P-hard

While reading an answer by Peter Shor and an earlier question by Adam Crume I realized that I have some misconceptions about what it means to be $\mathsf{P}$-hard. A problem is $\mathsf{P}$-hard if ...
Artem Kaznatcheev's user avatar
22 votes
3 answers
1k views

Limits to Parallel Computing

I am curious in a broad sense about what is known about parallelizing algorithms in P. I found the following wikipedia article about the subject: http://en.wikipedia.org/wiki/NC_%28complexity%29 The ...
Vladimir Levin's user avatar
22 votes
1 answer
1k views

Problems in BQP but conjectured to be outside P

Wikipedia listed four problems that are in $BQP$ but conjectured to be outside $P$: Integer factorization; Discrete logarithm; Simulation of quantum systems; Computing the Jones polynomial at certain ...
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