Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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3 votes
0 answers
115 views

Complete problems for $(NP\cap CoNP)/poly$ class and universal representations

It is conjectured that $NP\cap CoNP$ does not have a complete problem with respect to the polynomial-time many-to-one reductions. I would like to know the current knowledge about the nonuniform ...
3 votes
1 answer
315 views

Are there two definitions of Cobham's thesis?

In wikipedia, Cobham's thesis (or Cobham-Edmonds thesis) states: computational problems can be feasibly computed on some computational device only if they can be computed in polynomial time So ...
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4 votes
0 answers
198 views

What does width $4$ permutation branching program correspond to?

$L$ can be computed by a family of programs over $S_3$ of polynomial length if and only if $L$ can be computed by a family of $MOD3 ◦ MOD2$ circuits of polynomial size. $L$ can be computed by a family ...
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2 votes
0 answers
211 views

Simultaneous evidence for $L\neq NL$ and $P\neq NP$

We believe $L\neq NL$ and $P\neq NP$. Is there any evidence which simultaneously imply $L\neq NL$ and $P\neq NP$?
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3 votes
0 answers
83 views

Useful notion of ambiguous growing context-sensitive language

As far as I understand there is no useful notion of ambiguous context-sensitive language. For example for any inherently ambiguous context-free language there is a context-sensitive grammar generating ...
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2 votes
0 answers
413 views

The implication of $ S(SAT)=2^{\Omega{(n)}} $ Conjecture (5.7 in Wigderson Book)

I started to read Avi Wigderson book Math and Computation, which get me excited about the following: Conjecture 5.7. $ S(SAT)=2^{\Omega{(n)}} $, where $S$ denotes the size of the smallest Boolean ...
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2 votes
1 answer
74 views

Relative error estimation of a special type of GapP function

Consider the functions included in the complexity class GapP. We know that approximating a function from GapP, in the worst case, to inverse polynomial multiplicative error, is #P-hard. Even correctly ...
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3 votes
1 answer
107 views

Hardness when restricted to an infinite number of far apart instance sizes

Is there a result that rules out (under common complexity theoretic assumptions) that one can solve an NP-hard problem in polynomial time for an infinite number of possibly very far apart instance ...
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1 vote
0 answers
230 views

An $O(n^2\log^c{n})$ algorithm for matrix multiplication in this paper?

In the newest version of this paper by Yijie Han, the author claims that matrix multiplication can be solved by an $\tilde{O}(n^2)$ algorithm. It should be a big result, but it is still in arxiv and ...
3 votes
1 answer
280 views

Proof and computational complexity

I couldn't find documents elaborating on this: if the Curry Howard correspondence is to be interpreted as establishing a strong relation between proofs and programs, should there not be a strong ...
4 votes
1 answer
232 views

Complexity of relaxed edge colouring

A (proper) $k$-edge colouring of a graph $G(V,E)$ is a function $f:E\to\{1,2,\dots,k\}$ such that adjacent vertices are mapped to different colours; that is, $f(e)\neq f(e')$ if $e$ and $e'$ are ...
1 vote
1 answer
198 views

KRW Conjecture: separation of NC^1 and P

More than a real question this is a recap of something I have been studying. I hope someone will help me getting things straight, so any correction or thought about the following reasoning is more ...
0 votes
1 answer
71 views

Given a partition and an element, find the subset that includes this element

I am interested in the following simple problem: Let $X$ be a set and $X_1\cup X_2\cup\cdots\cup X_k$ be a finite partition of $X$. Given $x\in X$, find the subset $X_i$ for which $x\in X_i$. I am ...
1 vote
0 answers
216 views

anything hinting that EXPTIME $\subseteqq$ PSPACE?

Anything or evidence hinting that $$EXPTIME \subseteqq PSPACE$$?
4 votes
1 answer
517 views

Can we efficiently convert from NFA to smallest equivalent DFA?

Definitions For any automaton $X$, let $L(X)$ denote the language recognized by $X$. For any language $L$, let $sc(L)$ denote the number of states in the smallest DFA $X$ such that $L = L(X)$. ...
3 votes
3 answers
175 views

The complexity of LH with constant gap

Kitaev's quantum equivalent of the Cook-Levin Theorem, provides a polynomial time classical reduction from a QMA verification circuit to a sum $H$ of local hamiltonians, such that the least eigenvalue ...
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1 vote
0 answers
79 views

Additive error approximations of GapP functions

Consider a GapP function $g(x)$ for $x \in \{0, 1\}^{*}$. Consider an approximation $\tilde g(x)$ such that \begin{equation} \left|g(x) - \tilde g(x)\right| \leq \epsilon. \end{equation} Consider a ...
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2 votes
1 answer
219 views

DFSA and NFSA intersection problem

Given $k$ deterministic FSAs of $n$ states the intersection of their languages is empty is decidable in $n^{o(k)}$ time is an open problem. For unbounded $k$ it is known the problem is $PSPACE$ ...
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3 votes
0 answers
175 views

Improving the approximation in Stockmeyer's counting theorem

Given a $\#P$ function $f(x)$, we can use Stockmeyer's counting theorem to get an approximation $g(x)$ such that \begin{equation} \left(1 - \frac{1}{\text{poly}(n)} \right) f(x) \le g(x) \le \left(...
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1 vote
1 answer
56 views

Amortized time and worst case (non-amortized) separation

Assume a reasonable computation model (thinking about pointer machine or RAM model), is there a problem where there is a clear separation between amortized and worst case complexity? Say, if ...
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0 votes
0 answers
209 views

$CH=UL$ and partial breaking of transitive closure bottleneck problem and Savitch's theorem?

Let $L^t=DSPACE[O(\log n)^t]$, $NL^t=NSPACE[O(\log n)^t]$ and $UL^t=USPACE[O(\log n)^t$. Savitch provides $NL\subseteq L^{2}$. If $CH=UL$ we clearly got rid of the transitive closure bottleneck ...
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1 vote
0 answers
46 views

A proved computationally-irreductible function

In arXiv:1111.4121 and arXiv:1304.5247, Hervé Zwirn and Jean-Paul Delahaye propose a formal definition of computational irreducibility. Is there a (possibly artificial) function that can be proved to ...
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3 votes
1 answer
133 views

What is the best simulation of majority utilizing $\bmod\{2,3,\dots,p\}$ gates?

It is known $AC^0[2]$ cannot get majority function. Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$? What is ...
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4 votes
1 answer
210 views

Is $GCT$ necessarily a negative result program?

$GCT$ is a candidate program to separate permanent and determinant through symmetries. If indeed permanent and determinant can be handled in similar complexity class would $GCT$ be a program which can ...
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1 vote
0 answers
132 views

Algorithms for finding unique solutions of NP-complete problems

The complexity of algorithms that find unique solution for an NP-complete problem (the input is guaranteed to have a unique solution) seems to shed light on the hardness character of different NP ...
1 vote
1 answer
218 views

Pursuing Theoretical Computer Science after CS major

So I am currently a sophomore majoring in Computer Science. In the Data Structures course that I am currently studying, I studied the basics of complexity of a program and big O-notation, etc. That ...
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9 votes
1 answer
221 views

Is the Triangle Finding decision problem in $coNTIME(\tilde{O}(n^2))$?

The Triangle Finding decision problem asks whether there exists a triangle in a graph $G$ containing $n$ vertices. A triangle is a triple of vertices $(a, b, c)$ such that $a$ is adjacent to $b$, $b$ ...
4 votes
0 answers
169 views

The graph of problem reductions

A classical approach to study the complexity of a problem $P$ is to efficiently reduce a well known problem $P'$ to $P$, thus showing that $P$ is at least as difficult as $P'$. The TCS literature ...
3 votes
0 answers
173 views

PSPACE-complete under NP reduction

Is there some example of a PSPACE problem that we can show PSPACE-hard under NP reduction, but we do not know a proof of PSPACE-hardness under P reduction ? To be more precise, the NP reduction I am ...
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0 votes
1 answer
165 views

Comparing SAT to MCSP reduction class separations and faster SAT class separations?

Assume $SAT$ is in $QuasiP$. We immediately infer $NQuasiP=QuasiP$ and $EXP=NEXP$. From https://people.csail.mit.edu/rrw/easy-witness-nqp.pdf we infer $NQuasiP$ is not in $P/poly$ which implies $...
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1 vote
0 answers
30 views

Clauses structure as quenched random matrix for random $k$-SAT problems

In the Section III of Statistical mechanics of the random K-SAT model, the clauses structure of random $k$-SAT problems are expressed as $M \times N$ quenched random matrix where the numbers of ...
4 votes
0 answers
113 views

Relationship between SC and NL

It is a major open problem whether $NL \subseteq SC$, or equivalently, whether directed reachability can be solved (simultaneously) in poly-logarithmic space and polynomial time. What is known ...
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3 votes
2 answers
146 views

Looking for some lecture videos on logic, models of computation and computational complexity/tcs fundamentals [closed]

Looking for some lecture videos (introductory level) on logic, models of computation as well as computational complexity/ other theoretical computer science fundamentals
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0 votes
0 answers
118 views

$\mathsf{ACC}^0$ and $\mathsf{TC}^0$ with $\mathsf{Cuniform}$-$\oplus\mathsf{L}$ or $\mathsf{Cuniform}$-$\mathsf{NC}^1$ oracle?

$\mathsf{TC}^0$ is a small class with $\oplus\mathsf{L}$ containing it. Following inclusions are known: $$\mathsf{Cuniform}\mbox{ -}\mathsf{ACC}^0\subseteq\mathsf{Cuniform}\mbox{ -}\mathsf{TC}^0\...
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3 votes
0 answers
102 views

Increasing Functions in Non-deterministic Time Hierarchy Theorems

I was going over the proofs of the non-deterministic time hierarchy theorem (the one in Arora-Barak and the one by Fortnow and Santhanam). They are available here: http://theory.cs.princeton.edu/...
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2 votes
1 answer
118 views

Is there any relation between $\mathsf{PPP}\subseteq\mathsf{TFNP}$ and $\mathsf{UP}$? [closed]

A complete problem for $\mathsf{PPP}$ is the pigeon problem which is 'Given a Boolean circuit ${\displaystyle C}$ having the same number ${\displaystyle n}$ of input bits as output bits, find either ...
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5 votes
1 answer
258 views

Is there a containment between $\mathsf{ZPP}$ and $\mathsf{UP}$?

Do we know if $\mathsf{ZPP}\subseteq\mathsf{UP}$ known or is there oracles against the hypothesis?
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33 votes
12 answers
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,...
2 votes
0 answers
91 views

Does the awards budget cut problem support a sub $O(n\log n)$ time solution?

There's a famous problem these days in the interview prep community (particularly in PRAMP) called the awards budget cut problem. The problem gives you an input of $n$ integers called grants $g_1 ... ...
1 vote
0 answers
54 views

Complexity Lower Bounds for 3D Sparse Gaussian Elimination

I'm interested in lower bounds on the complexity in the real-RAM model of solving systems of linear equations which have the sparsity pattern of a three-dimensional cubic mesh. Specifically, consider ...
-4 votes
1 answer
27 views

Help find algorithm for array-based task

Given array if numbers a[1..n]. Pair of numbers (i, j) is interesting, if i < j и a[i] > 2a[j]. How to count number of interesting pairs in O(nlogn)? What is the solution? My solution is not ...
12 votes
0 answers
352 views

NP complete problem help

I'm currently trying to find a reduction to this problem: Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...
1 vote
0 answers
74 views

Complexity class name for the class of languages that are $\Sigma^1_1$-definable over finite domains

Let ${\cal L}=\{Y_1,..., Y_k, X\}$ be a finite relational language such that $X$ is a unary relation name. Let $\phi(X,\bar{Y})\in{\cal L}$ be a first-order formula (the formula can have the equality ...
1 vote
0 answers
216 views

Is there a known lower-bound on what the exponent could be, even if it turned out that P=NP?

Underlying motivation for the question: if someone showed that $\text{P}=\text{NP}$ but the algorithm thus produced for, e.g., $3\text{-SAT}$, runs in time $\Omega(n^G)$ where $G$ is Graham's number, ...
2 votes
0 answers
72 views

Computational complexity problem book with solution recommendation?

I will be taking complexity class next quarter and we will use the book "B. Barak, S. Arora, Computational Complexity: A Modern Approach". However, I have little exposure to complexity ...
10 votes
0 answers
169 views

Provable BPP Hierarchy

No Time Hierarchy theorem is known for $\mathsf{BPTIME}$, however, consider the following simple modification of the definition: A language is in $\mathsf{ProvableBPTIME}[f(n)]$ if there is a ...
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5 votes
2 answers
257 views

Can we recover integers $a_i$ from the sum $a_0 + a_1e+a_2e^2+\cdots+a_ne^n$?

Since $e$ is transcendental, the function $f:\mathbb Z_{\geq 0}^{n+1}\to \mathbb R$ is injective, $$ f(\underset{\text{Integers}\ \geq\ 0}{\underbrace{a_0,a_1,\ldots, a_n}}) = a_0 + a_1e+a_2e^2+\cdots ...
1 vote
0 answers
51 views

Living in Minicrypt, but sampling hard instances without the solution

In Impagliazzo's worlds, Minicrypt is the one, where one way functions exist. In other words, we can sample hard-on-average instances of NP complete problems. Question: Is living in Minicrypt, where ...
1 vote
1 answer
134 views

Complexity of a satisfiability problem

I would like the know the complexity of a specific satisfiability problem. I have a feeling it could be solved in polynomial time, but I am not sure about it. The problem is described below. Given $n$ ...
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2 votes
1 answer
136 views

Complexity of MAX-ONEs Monotone 2-SAT with $n^{3/2}$ or $C n^2$ clauses?

Let $\phi$ be negative monotone 2-CNF on $n$ variables and $n^{3/2}$ clauses. What is the complexity of finding satisfying assignment with maximum number of ones $k$? Alternatively let $G$ be a graph ...
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