Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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89 views

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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+100

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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1answer
250 views

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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68 views

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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1answer
92 views

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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52 views

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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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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1answer
83 views

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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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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1answer
153 views

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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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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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
148 views

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 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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2answers
224 views

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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1answer
135 views

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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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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122 views

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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1answer
152 views

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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79 views

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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72 views

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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221 views

$\#$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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3answers
121 views

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
278 views

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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1answer
54 views

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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222 views

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 ...
3
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103 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
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1answer
225 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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189 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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195 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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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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402 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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1answer
52 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 ...
3
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1answer
101 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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Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$

Crossposted at MathOverflow A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$. $\mathbf{1}$ is ...
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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 ...
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1answer
255 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 ...
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1answer
214 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 ...
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1answer
148 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 ...
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1answer
70 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 ...
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203 views

anything hinting that EXPTIME $\subseteqq$ PSPACE?

Anything or evidence hinting that $$EXPTIME \subseteqq PSPACE$$?
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97 views

Monomial Sparsity of Boolean Functions

Suppose you have some boolean function $f: \{-1,1\}^n \rightarrow \{-1,1\}$ with rational coefficients such that all degree 1 monomials of $f$ have a nonzero coefficient and the degree $n$ monomial ...
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0answers
48 views

Complexity of finding typing derivation trees

In type theories where type checking is decidable do we have estimates for how much time/space it takes to find a typing derivation tree of a valid typing judgment? Do any published references do this ...
4
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1answer
361 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 $L = L(X)$. ...
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3answers
151 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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66 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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