Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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Recognition problem of cycle permutation graphs

A cycle permutation graph is a cubic graph composed from two $n$-cycles each labeled $1, 2,..., n$, with additional edges connecting vertex $i$ in the first cycle to vertex $\pi(i)$ in the second, ...
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Problems in dynamic algorithms in computational geometry

The publication of Chiang and Tamassia's paper on dynamic algorithms in computational geometry included several algorithms used in solving dynamic computational geometry problems such as: Dynamic ...
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Is there a reason we haven't been able to prove that the existence of natural NPI problems even conditionally under assumption NPI is not empty?

We can write ${\mathsf {NP}}-{\mathsf P}= {\mathsf {NPC}}\cup {\mathsf {NPI}}$ where ${\mathsf {NPC}}$ is the set of ${\mathsf {NP}}$-complete languages (not in ${\mathsf {P}}$ by this partition), ...
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How many exponentiations can the Shamir algorithm reduce?

The Shamir's algorithm is depicted as follows (cited from Handbook of Applied Cryptography, chapter 14, algorithm 14.88, page 618) Shamir's algorithm INPUT: group elements $g_0,g_1,\dots, g_{k-1}$ ...
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Complexity of edge coloring graphs with $\Delta(G) \ge n/3$

Let $C$ be the class of graphs satisfying $\Delta(G) \ge n/3$, where $\Delta(G)$ is the maximum degree of a graph $G$, and $n$ denotes the number of vertices. What is the complexity of edge coloring ...
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240 views

Notion of associativity for ternary relations?

Consider the problem $\def\GEN{\mathrm{GEN}}\GEN$: given a ternary relation $R \subseteq X^3$ and a subset $S \subseteq X$, is the closure of $S$ with respect to $R$ equal to the whole set $X$? By ...
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Complexity of a variant of matrix multiplication

Assume a family of $n\times n$ integer matrices $\{M_a \mid a\in A\}$ for some finite set $A$, I want to decide whether for given vectors $\alpha$ and $\beta$, $\alpha M(w) \beta=0$ for all $w\in A^{...
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Games where $\omega(G) < \omega^*(G) < \omega^{ns}(G) < 1$?

A two player game $G = (I,O,V,p)$ is such that, if two non-communicating players Alice and Bob are given questions $(x,y)\in I^2$ drawn from the probability distribution $p$, they are supposed to ...
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232 views

The weakly NP-complete problems and their associated counting problem

Are there weakly NP-complete problems whose associated counting problem can be computed in pseudo-polynomial time? And if one were to be found (and assuming it is #P-complete), what would be the ...
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120 views

What is the complexity of decomposing 2k-regular multigraphs into k-stars?

Imagine we have a graph, which is 2k-regular, and we want to decompose it into copies of k-stars. Then we could just at first decompose the graph into 2-factors, orient each of the factors and then we ...
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468 views

What is the complexity relationship between counting and enumeration problems?

Thanks for the comments, I refined the question. What is the complexity relationship between counting and enumeration problems? If a counting problem is #P-complete, it means that the enumeration ...
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173 views

What do we know about $\text{P}^\text{NE}$

I have a $\text{NEXP}$-hard problem, that can be solved by a $\text{NEXP}^\text{NP}$ algorithm using a single oracle call. So from Hemaspaandra we know it is in $\text{P}^\text{NE}$, giving us $\text{...
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The number of maximal subsets with sum less than $m$

I've met this problem. I would like to know to which complexity class it belongs. Input a set of positive integers $I$, an integer $m$, an integer $n$. Question Is the number of $S \subseteq I$ ...
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149 views

Vertex Covers whose vertex induced subgraph has an even number of edges and no isolated vertices

Let $G$ be a graph, and let $C_{E,0}$ be the number of those vertex covers of $G$ satisfying both the following properties: Their corresponding vertex induced subgraph has an even number of edges. ...
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269 views

What does “no integrality gap” imply?

I'm currently working on a linear time heuristic for the rectangle decomposition of a binary matrix. This problem has a polynomial time solution, which in our case is too slow for large-scale ...
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What is the query and randomness complexity for very efficient PCPs?

In the 2012 paper On the Concrete-Efficiency Threshold of Probabilistically-Checkable Proofs, the authors state the following (paraphrased from page 11). Theorem 1 (informal). There is a PCP system ...
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552 views

Nondeterministic linear time vs. the deterministic time hierarchy

How much is known about nondeterministic linear time? I'm aware that $$ \mathrm{NTIME}(n) \neq \mathrm{DTIME}(n).$$ Is there an $m > 1$ so that $\mathrm{NTIME}(n) \not\subset \mathrm{DTIME}(n^m)$? ...
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Can relativization change the direction of separation?

Are any $A$, $B$, and $O$ such that: $O$ is a set (for oracle), $A$ and $B$ are the names of two known complexity classes, $A^X$ and $B^X$ have well-defined accepted meanings, $A=A^\emptyset\subset B^...
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Real-time countable vs fully time-constructible

Real-time countable functions were used in time hierarchy theorem in the papers of Hartmanis and Stearns (Theorem 9, 9.1 ...) and also of Hennie and Stearns (Theorems 3, 5, 7 ...). Now it is a "...
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Are there applications of experimental mathematics in TCS?

In recent years there have been major, diverse, sometimes surprising advances in experimental mathematics [1] for a variety of sophisticated uses such as developing/deriving exact formulas, theorem ...
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174 views

Complexity of computing logarithm of a prime power

Suppose $n = p^k$ for some prime number $p$ and some non-negative integer $k$. What is (the best-known upper bound on) the complexity of computing $k$ on input $n$ (given in binary)? It is important ...
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A natural result that relativized to a random oracle is true with probability 1/2

There are several well known results regarding random oracles, e.g. $\mathsf{IP}^A \neq \mathsf{PSpace}^A$ for almost all oracles. Are there any known natural examples where a similar statement ...
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150 views

How is Cooks overlap argument applied in Vitányi's theorem?

In P.M.B. Vitányi, Relativized Obliviousness, MFCS'80 paper one can read that the proof of Theorem 1 is based on the overlap-argument of Cook, however I don't see how this argument is applied. The ...
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How to prove deg(f) = n iff the parity imbalance of f is non-zero?

Not sure if the notation I'm using here is standard or not. I'm going over class notes and I'm stumped over an exercise given: Show that $deg(f) = n \iff PI(f) \neq 0$. Here $f$ is a boolean function ...
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Polytime Computable Distribution vs Polytime Sampleable Distribution

Many texts on average case complexity claim: 1) Any polytime computable distribution is also polytime sampleable. 2) A polytime sampleable distribution is one where a polytime probabilistic TM ...
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Evaluations of arithmetic circuits

Assume that $C$ is an arithmetic circuit of size $s$ that computes function $f(\vec{x},\vec{y})\in C[x_1,x_2,\ldots,x_d,y_1\ldots y_n]$. Does it always possible to find a basis $\vec{x}_1,\vec{x}_2,\...
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Problem with bitwise AND/OR on binary strings

Let $W = \{s_1, s_2, \ldots, s_n\}$ be a set of $m$-length binary strings and let $\land$ and $\lor$ be the binary operators of bitwise AND and OR. A bitwise formula $\phi$ is composed of operands in $...
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A approximation version of the Goldreich-Levin Theorem

A little introduction The Goldreich-Levin Theorem says that let $f$ a one-way function and set $f'(x,r)=(f(x),r)$ where $|r|=|x|$ then $\langle x, r \rangle = \sum_{i}x_ir_i \mod 2$ is an ...
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Question on size of words in Vandermonde Matrix - Vector multiplication complexity

I am trying to understand how word sizes in a problem affects complexity. The question could be a simple technicality I am trying to clarify since I am not from mainstream CS. Let $V$ be an $n \times ...
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Integral k-multicommodity flow with demands on acyclic digraphs wirh maximum outdegree two

It is well-known that different variants of Multicommodity flow problem are NP-complete. What is the complexity of the following variant, that is, the integral k-multicommodity flow problem with ...
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271 views

What is the complexity of #satisfiable instances of k-SAT ?

To continue the question posted by user1749 on Oct 13 2010 : How many instances of 3-SAT are satisfiable? Which was: Consider the 3-SAT problem on n variables. The number of possible distinct ...
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Complexity of Max Bisection on cubic planar graphs?

Max Bisection problem is to partition the set of nodes into two equal size sets such that the number of crossing edges is maximum. Max Bisection is $NP$-complete on cubic graphs and also on planar ...
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282 views

Is it known if $CFL \subseteq NSPACE(o(log^2(n)))$?

$CFL$ is the class of context-free languages. Question Is $CFL$ known to be solvable in $o(log^{2}(n))$ non-deterministic space? What about $DCFL$?
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Example of a hardness-of-approximation proof which improves the approximation factor?

Are there any examples of hardness-of-approximation reductions where we get a better hardness bound for the problem we've reduced to than the problem we've reduced from? In the examples I've seen so ...
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Problem of determining if a $4$ connected graph has $k$ Hamiltonian cycles

Definition: Define the $k$-HamiltonianCycles problem as the decision problem that asks if a given graph has at least $k$ distinct Hamiltonian cycles. Question: Is there some constant $k$ so that the $...
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Counting quotient graphs, but not exactly

All graphs considered will be directed graphs $G=(V,E)$, with $E \subseteq V \times V$ (so possibly with self-loops). For $k \in \mathbb{N}_{\geq 1}$, I will write $[k]$ the set $\{1,\ldots,k\}$. A $k$...
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Take a NEXP-complete problem and then have the input in unary. Why is this not NP-complete?

It is known that if any unary language is NP-complete, then P=NP. Suppose we take a NEXP-complete language with input $x$ in binary and witness $y\in\{0,1\}^{2^{poly(|x|)}}$ such that the verifying ...
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Complexity of counting Wang tiles

Consider the question of counting Wang tilings on a torus. The decision version of this problem is known to be NP-complete. Is the counting version #P-complete?
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Is #PP2DNF hard to approximate?

The problem #PP2DNF asks to count the number of satisfying assignments of a positive partitioned 2-DNF Boolean formula, i.e., a formula $\phi$ on variables $X_1, \ldots, X_n, Y_1, \ldots, Y_m$ of the ...
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Pseudodeterministically choosing elements from efficiently samplable distributions (or, the plausibility of a weak choice principle)

Suppose we have a poly-time samplable family of distribution. I.e., a family of distributions $D_n \subseteq \{0, 1\}^{\mathsf{poly}(n)}$ and an algorithm $S$ for which $D_n = (r \leftarrow^\$ \{0,1\}^...
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207 views

What is consequence of $PH\subseteq NSPACE((\log n)^2)$?

What is consequences of $PH\subseteq NSPACE((\log n)^2)$? We don't even know PH is equals to L or not. I am wondering what will be happened when $PH\subseteq NSPACE((\log n)^2)$?
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Critical Assignments vs Read-Once Branching Programs - Reference Request

Straight to the point: I'm looking for a reference for the fact that the complexity of a read-once branching program solving the search problem for an unsatisfiable formula $F$ is at least the ...
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Implications of resolving $BPP$ vs $PSPACE$

The relationship between the complexity classes $BPP$, $P$, and $NEXP$ is currently undetermined. We know that $P \neq EXP$ by the time hierarchy theorem, but we don't know if $BPP = P$ (as many ...
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133 views

Problems complete for non-deterministic PSPACE

We know that PSPACE = NPSPACE by savitch's theorem , but before that was proved , what problems were known to be complete for NPSPACE ?
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What are the consequences if $W[i]=W[i-1]$?

$FPT=W[1]$ does not collapse the $W$ hierarchy however falsifies $ETH$ belief. Is there non-trivial consequence if $W[i]=W[i-1]$ and any other consequence at $W[1]$?
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PTIME or NP-Hardness of stochastic objective function

I will begin by linking a previous post where I asked a general question for a stochastic setting which I describe below. It turns out that my "proof" for a restricted case had a mistake and there is ...
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Showing hardness of maximizing stochastic objective function over graph

Consider a graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each vertex $v_i$ can take positive value $a_i$ with probability $p_i$ and value $0$ with probability $1-p_i$. The challenge is to ...
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Hardness of Approximation for minimum path cover in an undirected graph?

Given an undirected graph $G = (V,E)$, a path cover is a set of disjoint paths such that every vertex $v\in V$ belongs to exactly one path. The minimum path cover problem consists of finding a path ...
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Why are one way functions and pseudorandom number generators considered necessary or essential for derandomization?

If strong pseudorandom number generator exists then $BPP=P$ holds and if one way functions exists then $BPP\subseteq SUBEXP$ holds. What are the best statements we have proved that come close to ...
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274 views

Convention for RAM machine models

When algorithm asymptotic runtimes are given without explicitly noting the computational model, what is the convention for the exact model used? My understanding is that most problems use unit-cost ...