Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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Fastest Known Algorithm for $k$-Dimensional Matching and $k$-Exact Cover

Given a $k$-uniform hypergraph $G$ (i.e., each edge of $G$ contains precisely $k$ vertices) on $n$ vertices, the $k$-Exact Cover problem is the task of deciding if there exists $n/k$ edges in $G$ ...
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Derandomizing arbitrary width *read-many* and *ordered* branching programs?

Modifying following TedP We know that derandomizing width $5\leq k\in O(1)$ read many branching programs is equivalent to $BPNC^1=NC^1$ and derandomizing width $k\in\Omega(n)$ read once ordered ...
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Does this sequence of quantified boolean formulas make sense to anyone else? most of these qbfs are valid

do all logicians on earth know that a question is a qbf? yes and they know the answer to a qbf is always yes or no? yes ok good all logicians know that solving one qbf is hard and buggy? yes but ...
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Fine-Grained Hardness for Undirected Hamiltonicity

The fastest known algorithm for detecting Hamiltonian cycles in directed graphs on $n$ nodes runs in essentially $2^n\text{poly}(n)$ time. However, for undirected graphs on $n$ nodes, there is an ...
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1answer
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Cook inspiration for NP completeness

An academic descendant of Cook just lectured on NP completeness. He said that the idea came from a well-known theorem in first-order logic that talks about completeness of satisfiability for ...
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What would be the consequences of PH=PSPACE?

A recent question (see Consequences of NP=PSPACE) asked for the "nasty" consequences of $NP=PSPACE$. The answers list quite a few collapse consequences, including $NP=coNP$ and others, providing ...
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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 ...
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Are runtime bounds in P decidable? (answer: no)

The question asked is whether the following question is decidable: Problem  Given an integer $k$ and Turing machine $M$ promised to be in P, is the runtime of $M$ ${O}(n^k)$ with respect ...
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Is polynomial-time the same in all classical computational models?

There are many models of computability, all giving the same notion of 'computable function'. To pick a few examples: Turing machines (with variants: one-ended, two-ended, multiple tapes...) RAM ...
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How exactly does derandomization of BPP imply lower bounds on NEXP

In their paper Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds, Kabanets and Impagliazzo prove that if $BPP=P$ then either $(i) NEXP \nsubseteq P_{/poly}$ or $(ii)$ ...
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Diophantine equations with bounds on variables

Solving Diophantine equations is famously known to be undecidable. What about Diophantine equations to be solved over a finite domain? In particular, if I put an upper bound $k$ over the value of the ...
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complexity class of a function - linear combinations and reductions (Fermionant, immanant, $GL_n$ representations)

The fermionant is a matrix function from physics, which is indexed by a positive integer $k$: \begin{align} \operatorname{Ferm}_k(A) = \sum_{\lambda} d_{\lambda}^{(k)} \operatorname{Imm}_{\lambda^T}(A)...
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Is finding the shortest consistent term to fill a missing line in a truth table still NP-hard?

I understand the logic minimization problem is NP-hard when given the onset, since the last step is equivalent to set cover optimization. If instead we are given a partial truth table, and we just ...
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EXPSPACE-complete problems involving numbers

This is a subset of this question. I'm looking for EXPSPACE-complete problems, for using in a reduction, which involve numbers in some ways, since my target problem involves numbers and linear ...
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Is the Kolmogorov complexity of the truth tables of the halting problem known asymptotically?

Let $HALT_n$ denote the string of length $2^n$ corresponding to the truth table of the halting problem for inputs of length $n$. If the sequence of Kolmogorov complexities $K(HALT_n)$ were $O(1)$, ...
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Research regarding human vs algorithm on NP-hard problems

I am interested in the comparison of human performance vs algorithmic performance on NP-hard problems, both with regard to exact solutions and approximations, with the intent to find areas where ...
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A query regarding relation b/w complexity classes $NP$ and $BQP$ in case of collapse of $PH$?

Assuming if $PSPACE$ collapses to the class $NP^{NP}$ $\mathsf{\implies PH=\Sigma_2^P}=\mathsf{\Pi_2^P = PSPACE}$. Since $\mathsf{NP\subseteq NP^{NP}}$ and $\mathsf{BQP\subseteq QMA \subseteq PSPACE} \...
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Asymptotic complexity lower bounds of proof checking

This paper on universal search mentions (on pp. 6-7) that proof checking can be done in $O(n^2)$ where $n$ is the length of the proof. Is this optimal? I don't want to specify the problem too ...
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Problems Between P and NPC

Factoring and graph isomorphism are problems in NP that are not known to be in P nor to be NP-Complete. What are some other (sufficiently different) natural problems that share this property? ...
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1answer
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Does Horn SAT (Horn formula in CNF) have an integral polytope?

In some ways, my question is related to this: Is the matching polytope integral? Matching and Horn-SAT are both polynomial time solvable.. So I wonder if there is a Horn polytope, similar to the ...
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minimizing size of regular expression for finite sets

It is known that minimizing the size of a regular expression is PSPACE-complete even if we have a DFA as the language's specification. What are the results if the language is finite? One can ...
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1answer
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Sorting with an average of $\mathrm{lg}(n!)+o(n)$ comparisons

Is there a comparison-based sorting algorithm that uses an average of $\mathrm{lg}(n!)+o(n)$ comparisons? Existence of a worst-case $\mathrm{lg}(n!)+o(n)$ comparison algorithm is an open problem (...
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Useful primitives CPUs could provide, from TC0 (or NC1)

I just listened to XYZ talking about "How Universal Is the Idea of Numbers?", and bashing the concept as an accidental historical artifact. He suggested that totally different computational ...
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1answer
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3SAT to 1-in-3SAT reduction with additonal constraints

The simplest Reduction for 3-SAT to 1-in-3-SAT reduction is as follows: For each 3SAT clause: $x+y+z=1$ Introduce 4 new variables $\{a, b, c, d\}$ and replace original clause with below 3 clauses: $R(...
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Is it known if $\mathrm{CFL} \subseteq\mathrm{ NSPACE}(o(log^2(n)))$?

$\mathrm{CFL}$ is the class of context-free languages. Question Is $\mathrm{CFL}$ known to be solvable in $o(log^{2}(n))$ non-deterministic space? What about $\mathrm{DCFL}$?
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1answer
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Is QMA known to contain Co-NP?

Is QMA known to contain Co-NP? If not, would Co-NP being contained in QMA have any implications for other complexity classes. (e.g. Causing the polynomial heirachy to collapse.)
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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, ...
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Complexity of "can we get a cycle by stacking directed bipartite graphs?"

Preliminaries We consider directed bipartite graphs of the form $G = (V,V',E)$, in which the nodes are partitioned into $V = \{1,\ldots,n\}$ and $V'=\{1',\ldots,n'\}$, with $|V|=|V'|=n$, and $E\...
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3answers
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Is Descriptive Complexity dead?

I recently started reading about Descriptive Complexity, the branch of Complexity Theory studying the logic languages needed to express complexity classes. The main milestone in the area seems to be ...
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1answer
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Minimum cut with size bounds $k\leq |S| \leq |V|-k$

It is known by the max flow min cut theorem that the minimum cut problem is in $P$. I am interested in knowing what is known on the complexity of the minimum cut with size $k\leq |S| \leq , |V|- k$. ...
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Questions about the equivalence between PH and depth-d circuits with respect to an oracle

Consider an oracle $A$ and the language \begin{equation} P(A) = \{x\in \{0, 1\}^{n}: \text{the number of strings in }~A~\text{of length}~|x|~\text{is odd}\}. \end{equation} I am trying to make sense ...
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1answer
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Graph classes where giving a q-clique edge cover makes testing for q-colouring easy

A $q$-clique of a graph is a complete subgraph on $q$ vertices. A $q$-clique edge cover of $G$ is a set of subgraphs of $G$ such that each subgraph is a $q$-clique and each edge of G is contained in ...
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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". ...
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An NP-hard Hidden Subgroup Problem

I've encountered a model which can be thought of as a version of the Hidden Subgroup Problem (https://en.wikipedia.org/wiki/Hidden_subgroup_problem), but that doesn't quite meet the standard problem's ...
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6answers
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Theoretical explanations for practical success of SAT solvers?

What theoretical explanations are there for the practical success of SAT solvers, and can someone give a "wikipedia-style" overview and explanation tying them all together? By analogy, the smoothed ...
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$\#$P hardness of computing weighted sum of degree $2$ polynomials

Consider polynomials $f: \{0, 1\}^{n} \rightarrow \{0, 1\}$ over $\mathbb{F}_2$ (addition and multiplication are taken modulo $2$.) Consider integers $x \in \{0,1\}^{n}$, written in binary. Let $\...
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Result showing that DTIME[T] is strictly contained in random-access NTIME[T]

I recall seeing a result showing that multi-tape DTIME[T] is strictly contained in random-access NTIME[T] for reasonably large T (so not the PPST proof with the $\log^\star$ sort of factors), but I ...
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Parametrization of context-sensitive language in polynomial time

Let $\Sigma$ be a finite alphabet. Let $L\subset \Sigma^*$ be a context-sensitive language containing a word of every length. Can we always find $f:\Sigma^*\to L$ computable in polynomial time in ...
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1answer
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Hardness of maximizing $x^TAy$ with $\{-1,1\}$ entries

My question concerns the NP-hardness of the following discrete optimization problem: Given a matrix $A \in \{ \pm 1 \}^{m\times n}$, $$\begin{array}{ll} \underset{x \in \{ \pm 1 \}^m ,\, y \in \{ \pm ...
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3answers
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Cost of Numerically Solving a System of P polynomials, each of V variables, and degree D to a Specific Accuracy

Let there be a set of $P$ polynomial equations $f_j(x_1,x_2...x_V)=0$ where $1\leq j\leq P$. For each $f_j$ the coefficients are real and every variable goes up to degree $D$. It is also guaranteed ...
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1answer
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Is time-slot matchmaking NP-hard?

I have the following problem, which I intuitively expect to be NP-hard but cannot see how to write a reduction for: Givens: There is a fixed set of time slots, e.g. {9-10, 10-11, 11-12, 12-1, etc.}. ...
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How is a "low-degree polynomial" precisely defined in Algebrization?

I'm going through papers which present algebrization as a barrier and I'm trying to understand how "low-degree" polynomials are precisely defined, i.e. are they low with respect to the ...
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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 ...
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1answer
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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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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 ...
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How to prove that USTCONN requires logarithmic space?

USTCONN is the problem that requires deciding whether there is a path from the source vertex $s$ to the target vertex $t$ in a graph $G$, where these are all given as part of the input. Omer Reingold ...
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$CH=UL$ 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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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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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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Are there any implementations of Cook-Levin?

Does anyone know if there is an implementation of Cook-Levin? A program that gets an input like the following: $M$: the code of a machine model/simple program in a programming language like C, $\vec{...

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