# Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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### Is there a natural restriction of VO logic which captures P or NP?

The paper Lauri Hella and José María Turull-Torres, Computing queries with higher-order logics, TCS 355 197–214, 2006. doi: 10.1016/j.tcs.2006.01.009 proposes logic VO, variable-order logic. This ...
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### Communication complexity for deciding associativity

Let $S=${$0,...,n-1$} and $\circ : S \times S \rightarrow S$. I want to compute the communication complexity of deciding whether $\circ$ is associative. The model is the following. $\circ$ is given ...
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### Easy decision problem, hard search problem

Deciding whether a Nash equilibrium exists is easy (it always does); however, actually finding one is believed to be difficult (it is PPAD-Complete). What are some other examples of problems where ...
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### Is $AC^0$ with bounded fanout weaker than $AC^0$?

In the survey "Small Depth Quantum Circuits" by D. Bera, F. Green and S. Homer (p. 36 of ACM SIGACT News, June 2007 vol. 38, no. 2), I read the following sentence: The classical version of $QAC^0$ (...
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### What are the current best known upper and lower bounds on the (un)satisfiability threshold for random k-sat and/or 3-sat?

I would like to know the current state of the phase transition for random k-sat, given n variables and m clauses, what is the best known c=m/n for upper and lower bounds.
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### 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. ...
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### Cobham's Result on Efficient Computations

In the following paper: Alan Cobham (1965), "The intrinsic computational difficulty of functions", Proc. Logic, Methodology, and Philosophy of Science II, North Holland. Cobham defined the class P ...
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### Separating Logspace from Polynomial time

It is clear that any problem that is decidable in deterministic logspace ($L$) runs in at most polynomial time ($P$). There is a wealth of complexity classes between $L$ and $P$. Examples include $NL$,...
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### Alternative proofs of Schwartz–Zippel lemma

I'm only aware of two proofs of Schwartz–Zippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz. Are there any other ...
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### References on Circuit Lower Bounds

Preamble Interactive proof systems and Arthur-Merlin protocols were introduced by Goldwasser, Micali and Rackoff and Babai back in 1985. At first, it was thought that the former is more powerful than ...
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### Simple question about decision problems

(I am in the middle of my first theoretical cs course, so I apologize in advance for what is probably a stupid question.) So, we say that some language L is in P, which means that a Turing machine ...
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### Is there any sparse language known to be in NPI under the $P \neq NP$ assumption ?

I wonder to know wether there are sparse language (even constructed by delayed diagolanization) in NPI under the assumption that $P \neq NP$.
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### Is Gap-3SAT NP-complete even for 3CNF formulas where no pair of variables appears in significantly more clauses than the average?

In this question, a 3CNF formula means a CNF formula where each clause involves exactly three distinct variables. For a constant 0<s<1, Gap-3SATs is the following promise problem: Gap-3SATs ...
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### Separation of limited nondeterminism classes?

It is interesting to find the best lower bound on the number of nondeterministic bits needed to solve satisfiability problem. Let $\beta_k P$ be the class of problems solvable by a nondeterministic ...
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### What is the best way to get a close-to-fair coin toss from identical biased coins?

(Von Neumann gave an algorithm that simulates a fair coin given access to identical biased coins. The algorithm potentially requires an infinite number of coins (although in expectation, finitely many ...
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### Relativization with Respect to Non-Recursive Oracles

In the paper Relativizations of the P = ? NP Question, Baker et al. showed that there are relativized worlds in which either P = NP or P ≠ NP holds. All oracles in their settings were recursive sets. ...
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### Complexity of advice language?

Let $L$ be a language in P/poly. There is then a deterministic polynomial-time Turing machine $M$ with polynomial-sized advice that decides $L$. Consider the language $A(M)$ of all advice strings ...
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### Is there known any complexity class containing online counterparts of optimization problems?

Is there known any complexity class containing online counterparts of optimization problems? If not, then how such class can be defined? We know that many problems have their online version: e.g. ...
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### Complexity of finding vectors with optimal projection?

Input: a set $T$ of vectors $v_i=(x_i,y_i,z_i)$. Where $x_i,y_i,z_i$ are integers. Output: a subset of vectors $v_1,v_2,...,v_n$ with vector addition $m=\sum v_i$ such that the projection of $m$ on ...
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### What is known about the complexity of finding minimum circuits for SAT?

What is known about the complexity of finding minimal circuits that compute SAT up to length $n$? More formally: what is the complexity of a function which, given $1^{n}$ as input outputs a minimal ...
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### Complexity Classes for Cases Other Than “Worst Case”

Do we have complexity classes with respect to, say, average-case complexity? For instance, is there a (named) complexity class for problems which take expected polynomial time to decide? Another ...
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### Barriers and Monotone Circuit Complexity

Natural proofs is a barrier towards proving lower bounds on the circuit complexity of boolean functions. They do not directly imply any such barrier in proving lower bounds on the $monotone$ circuit ...
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### Exhausting Simulator of Zero-Knowledge Protocols in the Random Oracle Model

In a paper titled "On Deniability in the Common Reference String and Random Oracle Model," Rafael Pass writes: We note that when proving security according to the standard zero-knowledge definition ...
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### Understanding QMA

This question comes out of an answer Joe Fitzsimons gave to a different question. Most natural complexity classes have a one-line "intuitive description" that helps characterize core problems in that ...
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### 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 ...
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### Algorithms and computational complexity of clique and biclique covers

I've been reading a paper by a mathematical chemist. He proposes some indices to measure the complexity of molecules. From here on in, instead of molecules, think undirected connected graphs: a ...
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### Intractability of NP-complete problems as a principle of physics?

I'm always intrigued by the lack of numerical evidence from experimental mathematics for or against the P vs NP question. While the Riemann Hypothesis has some supporting evidence from numerical ...
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### Is embedding a solution feasible for SAT?

I am interested in "hard" individual instances of NP-complete problems. Ryan Williams discussed the SAT0 problem at Richard Lipton's blog. SAT0 asks whether a SAT instance has the specific solution ...
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### Do the proofs that permanent is not in uniform $\mathsf{TC^0}$ relativize?

This is a follow up to this question, and is related to this question of Shiva Kinali. It seems that the proofs in these papers (Allender, Caussinus-McKenzie-Therien-Vollmer, Koiran-Perifel) use ...
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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 ...
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### Are there any classes of functions which require provably different resources to compute versus computing their inverse?

Apologies in advance if this question is too simple. Basically, what I want to know is if there are any functions $f(x)$ with the following properties: Take $f_n(x)$ to be $f(x)$ when the domain and ...
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### Results showing existence/non-existence of finite graphs with specific computable properties imply certain complexity results

Are there any known results showing that existence (or non-existence) of finite graphs with specific computable properties imply certain complexity results (such as P = NP)? Here's one completely ...
Tree width measures how close a graph is to a tree. It is NP-hard to compute tree width. The best known approximation algorithm achieves $O(\sqrt{{\log}n})$ factor. Courcelle's theorem states that ...
Is there any relationship between the number of vertex covers of a graph $G$ and the permanent of $G$'s adjacency matrix?