Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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Implication of lower bounds in Boolean circuit to other models of computation

Suppose that one can prove that some hard function $f$ with $n$ bit input does not admit any Boolean circuit of size at most $n^t$. Then, how strong can we say about how hard $f$ is in other models? ...
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intuition that VP=?VNP is (not?) connected to P=?NP

recently there has been major progress into the VP=?VNP problem for algebraic circuits originated by Valiant, inspiring some optimistic outlook on its eventual or imminent resolution.[1] what is an ...
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Period of a Multivariable Function using Quantum Computing

consider a function $$f(x_1,x_2...x_n)$$ I am told it is possible to compute the period of the function as a vector $$<l_1,l_2...l_n>$$ where each l denotes the period of the function for ...
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Finding a cellular automaton that reproduces a finite-sized spacetime patch with reduced coloration

Note: This question is an extension of: Computational complexity of finding a (1D deterministic) CA that achieves a desired spacetime history "patch" after $t$ iterations, which Marzio De ...
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102 views

Universal predictor for $\mathsf{L}$-sequences in $\mathsf{P}$?

Consider any language $K$. Define $s(K) \in {\lbrace 0, 1 \rbrace}^\omega$ (an infinite sequence of bits) by the recursive formula $$s(K)_n=\chi_K(s(K)_{<n})$$ Here $\chi_K$ is the characteristic ...
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Smallest Nonuniform Complexity Classes including uniform-P

As we know, studiyng differences between uniform complexity and nonuniform complexity class is crucial. For example, P/poly is defined as challenges to derive a separation between P and NP, because ...
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128 views

Computational Complexity of RESTRICTED primality testing

Input: Any number $n \in \mathbb{Z}^+$ that can be represented in the form of $n = 2^a + b,\ |b|= c $. output: YES if $n$ is prime , else NO . Now, length of binary input is $\log(a) + O(1)$ which ...
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Local Decoding and Hardness Amplification

I'm reading Arora and Barak's book "Computational Complexity", Chapter 19.4 "Local Decoding and Hardness Amplification". And I'm confused with the following question: Let $f:\{0,1\}^n\rightarrow\{0,1\...
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Estimating the number of operations required to factorize a $b$-bit integer using GNFS

You may skip to question According to Wikipedia, the complexity of factorizing a number n using the general number field sieve is ("heuristically"): $ \exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\...
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648 views

“Exact” (not just the order) computational complexity of eigenvalue decomposition

Does anyone know what the eigenvalue decomposition of a general n x n complex matrix is? By complexity I mean the number of multiplication operations. I know from another question posted on this site ...
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202 views

Minimization on a binary matrix

Assume you are given a matrix $$ X= \begin{bmatrix} x_1^1 & x_1^2 & \dots & x_1^m \\ x_2^1 & x_2^2 & \dots & x_2^m \\ \vdots & \vdots & \ddots & \vdots \\ x_n^...
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374 views

Succinct representation of boolean functions

Let $f$ be a boolean function over $n$ variables $f: \{ 0, 1 \}^n \rightarrow \{ 0, 1 \}$. We are looking now for a representation of $f$ s.t. when given that representation and values $x_1, \ldots, ...
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389 views

Hardness of min-max problems

Consider the following min-max problem Given a graph $G=(V,E)$ and an integer $k \geq 0$, delete at most $k$ nodes in $G$ to maximize the size of the minimum dominating set in the residual graph. ...
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Problems or issues with a proposed circuit class?

I'm looking to use something close to the following as a definition for a circuit class. This is obviously semi-informal. I am curious if any one sees any potential problems with it, or where ...
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259 views

Separating the QIP hierarchy

Background: I'm a CS grad student. I've taken a course on computational complexity. Question: Can you suggest an introductory book on quantum computation, especially regarding the details of ...
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377 views

A simple NP-hard problem

I'm reading this paper, A simple NP-hard problem by Demontis, where he defines an interesting $NP$-hard set and finally he conclude with this theorem: Theorem: $P=NP$ if and only if it is possible ...
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What evidences are there that $PP$ is in $BQP$ and $PP$ is not in $BQP$?

Unlike hierarchy collapse arguments for classical complexity we have that quantum complexity is different. What evidences are there that $PP$ is in $BQP$? What evidences are there that $PP$ is not ...
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Why is it difficult for GCT to prove super quadratic lower bound?

We have a quadratic lower bound for the Permanent Determinant problem. Why is it difficult for GCT to improve it?
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Complete problems for dtime

$P$ is the polynomial time class defined as $$\cup_{i=0}^{\infty}DTIME(n^i).$$ which has linear programming as complete problem. Are there complete problems under linear time reductions within each $...
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Can you diagnolize without mentioning simulation?

Are there any known diagonalization proofs, of a language not being in some complexity class, which do not explicitly mention simulation? The standard diagnolization argument goes: here is a list of ...
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Encrypt an explicit circuit

Suppose we are given a circuit that are promised to be in a fixed class $\mathcal{C}$ (say AC^0). We want to “encapsulate” the circuit such that the resulting circuit computes the same function, while ...
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Are there complexity theory consequences of the collapse NEXP=EXP^NP?

It is clear that $NEXP\subseteq EXP^{NP}$, as a TM with exponential run time can simply query the NP oracle with an exponentially long query. However, it's not clear that the reverse $EXP^{NP}\...
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Randomized Reduction for Maximization Problem

I have two maximization problems $P_1$ and $P_2$ where the decision version $L_1 = \{(x, t) : \operatorname{Val}_1(x)\ge t\}$ of $P_1$ is $\mathsf{NP}$-complete. Let $f:P_1\to P_2$ be a randomized ...
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Volume computation of special polytopes

I'm interested in computing the volume of a special class of $\mathcal{H}$-polytopes and the complexity of doing so. I know that in general it is #P-hard to compute the volume of $\mathcal{H}$ -...
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Complexity of extending $P_4 $-partition of cubic graphs

This is a question I posted on MathOverflow before but never got an answer. I am cross-posting it here. Surprising phenomena occurs when we want to extend a partial solution of some easy problems. We ...
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119 views

What definition for $FPT$ algorithm for $KSUM$ gives $W[P]=FPT\implies KSUM$ is $FPT$?

In the definition on $KSUM$ problem we are given $n$ input integers and we have to decide if $K$ of them sum to $0$. $KSUM$ is $FPT$ if there is a $O(f(K)poly(n))$ algorithm for it. However Downey ...
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On PP in communication complexity

Aho says $D(f)=O(N(f)N(\overline f))$ where $D(f)$ is deterministic communication complexity and $N(f)$ is non-deterministic version. Do we know $PP(f)=\Omega(2^{(N(f)N(\overline f))^{O(1)}})$ or $...
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On $i.o.P/poly$?

Is $NEXP^{NP}\not\subseteq i.o.P/poly$? Is there any consequence if $NP$ or $PP$ is in $i.o.P/poly$? Showing $NEXP^{NP}\not\subseteq P/poly$ needs Karp-Lipton. What is the best $i.o.P/poly$ lower ...
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284 views

Can $f^{2^N}(x)$ be computed in polynomial time when $f$ is linear?

Linear functions: definition Let's define a linear function as one expressible as an untyped λ-calculus term with the added restriction that no lambda argument can be used twice. Linear functions: ...
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Permanent in Bounded error Quasi Poly time

Is there any consequence to complexity theory if Permanent has a BQP (classical quasipoly version of BPP)? Is there any consequence to complexity theory if Permanent has a QP (classical quasipoly ...
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120 views

Is this graph communication game known?

Let $X_m=[m]=\{0,1,\dots,m-1\}$ and let $Y_m=[2m]\setminus [m]$. Given is a complete bipartite graph $G_m$, with parts $X_m$ and $Y_m$ and edges $\{x,y\}$ for every $x\in X_m$ and $y\in Y_m$. Alice ...
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Identity testing of margins of boolean functions

Consider a boolean function $f(\vec{x},\vec{y})$, we define an (integer-valued) function $g(\vec{x})=\sum_{\vec{y}}f(\vec{x},\vec{y})$, i.e., $g$ can be considered as a (sort of) margin of $f$. The ...
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118 views

Arithmetic complexity of matrix powering with non-commutative entries

Assume $M\in\Bbb Z_{\geq0}[x_1,\dots,x_n]^{m\times m}$ be an $m\times m$ matrix in $n$ variables with $x_i$ being non-commutative. What is the complexity class and circuit and formula complexity of ...
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Complexity classes for problems that can be solved only from the length of the input

A tally language is a language on an alphabet with only one symbol. One can define complexity classes for tally languages, such as $P_1$ (the tally languages that can be decided in polynomial time). ...
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NP-hardness of minimizing sum of complicated objective function

In our research, we faced the following problem optimization problem: Input: a list of $k$ pairs of positive integers $(n_1,d_1), \ldots, (n_k,d_k)$; an integer $m$. Output: $P$, a partition of the ...
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Understanding MA protocol as a variant of TM for small space setting

MA protocol is one of the most basic models of interactive proofs. Merlin is a prover sending a witness $w$ for given input string $x$, and Arthur is a verifier who verifies if $w$ is a positive ...
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In what complexity classes other than $NP$ are these problems related to unary languages?

If I remember correctly saw this reduction in a paper can't find at the moment. Consider the following NP-complete variation of the Subset Sum problem. Given a set of positive integers $S=\{x_1,\...
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ExpSpace problems whose configuration reachability problems are in P/poly?

Is anything known about ExpSpace problems whose configuration reachability problems are in P/poly? Let $M$ be an ExpSpace machine. Given two configurations $a$ and $b$ of $M$ (of max length), ...
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Definition of convex optimization problem by Stephen Boyd and Lieven Vandenberghe

Boyd and Vandenberghe say that a convex optimization problem is one of the form: minimize $f_0(x)$ subject to $$f_i(x)\le 0, i=1,\ldots m$$ $$a_i^\top x=b_i, i=1,\ldots p$$ ...
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Turing reduction from integer factorization to clique

Is there a general web repository of reductions between and among various NP problems? In particular, I'm looking for a direct Turing reduction from integer factorization (candidate for NP-...
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How to determine if a function is negligible?

In cryptography (and probably in many other areas) there is a huge usage of negligible functions when proving theorems. Although I know what is a negligible function, every time I encounter a ...
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Determining if a function is constant or not using period finding

Consider an arbitrary boolean function $$f: {\lbrace 0,1 \rbrace}^n \rightarrow \lbrace 0,1 \rbrace$$ which we write as: $$f(x_1, x_2 ... x_n) $$ where each $x_i$ is a boolean variable We note ...
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Computing the period of a function using a quantum computer

Consider a blackbox function $$f(x): Z \rightarrow \lbrace 0,1 \rbrace $$ Which inputs an integer and outputs 0 or 1 with bit complexity n. If the period P of this function satisfies $$P \in O(2^{...
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Significance of Logic in Computer Science

I understand the significance of the theory of comptuation, for example NP-hardness of a problem signals us to forget about implementing it's exact solution and rather try approximating it. In the ...
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Extended Formulaiton and Integer Programming

An extended formulation (EF) of a polytope $P\subseteq \mathbb{R}^d$ is a system of linear constraints $Ex + Fy = g, y\geq 0$ in variables $(x,y)\in \mathbb{R}^{d+r}$ where $E,F$ are real matrices ...
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Nonmetric TSP and Functional Compleixty Classes

Non-metric TSP that is TSP and some instance is not hold the triangle inequality is NP-hard by gap-reduction method. Is this general TSP a complete problem in some functional complexity classes ? I'...
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Complexity of the packing

Let $(A, \leq)$ be a totally ordered alphabet. The packing ${\tt pack(u)}$ of a word $u \in A^*$ is the word of $\lbrace 1, \dots, k \rbrace^*$, where $k$ is the number of different letters of $u$, ...
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324 views

On Vertex Coloring of Permutation Graph and Comparability Graph and 2-SAT

I have 2 questions. Firstly, I am not sure about differences between Permutaion Graphs and Comparability Graphs. The latter graph class includes the other class. Is there a specific example of graph ...
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358 views

Maximize Covering Minimizing the Overlap

I have this problem: Given a collection of sets $S:\{S_{1},...,S_{k}\}$ where each set $S_{j}$ is a subset of $U:\{e_{1},...,e_{n}\}$ universe of elements. I would find-out a subset $C \subseteq S$ ...
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Is there a series of algorithms for approximating TSP polynomially?

I've began studying some CS recently, and I've faced the TSP. The decision problem version of the TSP is NP-complete, right? I've noticed (and elaborated myself) that there exists several polynomial ...

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