Questions tagged [counting-complexity]
How hard is counting the number of solutions?
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Crafting ${NP}^{\#P}$-complete problems
Some related posts:
Is $coNP^{\#P}=NP^{\#P}=P^{\#P}$?
$\mathsf{NP^{PP}}$ vs $\mathsf{P^{PP}}$
I needed a complete problem for the class ${NP}^{\#P}$ for a reduction to show the hardness of some other ...
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Coefficients of a determinant of a matrix of univariate polynomials is in $GapL$
Given any matrix of univariate polynomials of degree $\leq n^{O(1)}$ then prove that the coefficent of $x^i$ in the determinant of the matrix is in $GapL$
Hint: Use Mahajan-Vinay's result of ...
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Computational complexity of finding the $n$th Dedekind Number
Recently, two independent groups of researchers exactly calculated the $9$th Dedekind Number (see e.g. Quanta). The $n$th Dedekind Number counts the number of antichains consisting of subsets of $\{1,...
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Do fast satisfiability algorithms imply fast algorithms for parity SAT?
$\oplus$SAT is the problem of deciding if the number of satisfying assignments to a CNF formula is odd (and is the standard complete problem for the class $\oplus$P, or Parity-P).
Suppose we have a ...
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FPRAS to estimate the probability to get a cyclic subgraph of a directed graph
Consider a directed graph $G = (V, E)$ whose edges are annotated with independent probabilities of existence. This gives a probability distribution on the subgraphs of $G$; for instance, if each edge ...
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Complexity of permanent verification
Consider the problem of permanent verification:
$\bullet \ $ Given a $n\times n$ matrix $A$ with entries in $\{0,1\}$, and given $k\ge 0$, does $Per(A)=k$?
Question: Is it known to be NP-hard? Should ...
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Complexity of "opposite" version of a variant of #Positive-2-SAT
In this post ,I introduced a new variant of #Positive-2-SAT .
This version of problem puts restrictions on the inputs of the
#Positive-2-SAT such that we can only choose at max only 2 clauses from ...
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Perm and Det mod $2^k$ - I
Given a $0/1$ square matrix, the permanent and determinant modulo $2^k$ is in $\oplus P$ and $\oplus L$ respectively for any fixed $k$. In fact both are in $\oplus L$ (in fact in $\oplus SPACE(k^2\log ...
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Extending fagin’s theorem for #P (for arbitary structure)
While i am reading Descriptive complexity of #P functions (Saluja) in theorem 1 he state that #FO coincides #P on ordered structures.
This is a corollary from fagin’s theorem. I have read fagin’s ...
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Complexity of sampling a clique uniformly at random
Let $G$ be an undirected graph, and let $C_1, ..., C_M$ denote all possible cliques in $G$.
What is known on the complexity of sampling a clique uniformly at random. That is, returning clique $C_i$ ...
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$\mathsf{coNP}^{\mathsf{\#P}}$ and $\mathsf{coNP}^{\mathsf{\#P}^\mathsf{\#P}}$
I was reading a paper that demonstrates that deciding whether a loop-free program is $\varepsilon$-differentially private is $\mathsf{coNP}^{\mathsf{\#P}}$-complete. What are some other problems that ...
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Complexity of the unique homomorphism problem up to automorphisms
I am interested in the following problem: given two relational structures $\mathbf{A},\mathbf{B}$,
is there a unique homomorphism from $\mathbf{A}$ to $\mathbf{B}$ up to automorphisms of $\mathbf{B}$, ...
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Number of equivalent formulas in a function-free first order logic language?
In this paper by Martin Grohe, in the first paragraph of section 4.1, it says:
"because upto logical equivalence there only exist finitely many $FO[\rho]$ formulas of quantifier rank at most $q$ ...
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Problems in $P^{PP}$
I just discovered that a problem that I was studying could belong to $P^{PP}$, I would like to prove that this problem is $P^{PP}$-complete (if that is even a thing). The issue is that I'm unable to ...
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Question about #P-completeness and NP-completeness
In the book Nature of Computation by Moore & Mertens there is an exercise saying "show that if a counting problem #A is #P-complete with respect to parsimonious reductions, that is if every ...
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Power of unique counting class
One question that recently encountered is the following, suppose I have a task $L$ which has input length $n$, the problem is in the $\text{NP}$ and I promise that there is a unique solution. (The ...
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On the proof of $PP = P \iff \#P = FP$ in Arora & Barak (Lemma 17.7)
Introduction
I am currently studying chapter 17 (of the famous textbook by Arora & Barak [1]) on the complexity of counting and got stuck on the proof of Lemma 17.7, which states $\mathrm{PP} = \...
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Approximate inclusion-Exclusion?
I am trying to understand or find literature on the following problem of approximate inclusion exclusion.
Let $S:=\{A_i\}_{i=1}^{m}$ be a set of $m$ sets. Every intersection of $k$ elements in $S$ ...
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Growth rate of Knapsack Solutions
Let's say I have a Knapsack with capacity $\tau$, and I have an infinite sequence of items with weights $(a_n)_{n=1}^{\infty}$. A feasible Knapsack solution is a subset $S \subset \mathbb{N}$ such ...
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Reducing counting minimal vertex covers to counting minimum cardinality vertex covers
Consider two problems.
Problem 1: Given a graph $G = (V, E)$, find the number of minimum cardinality vertex covers of $G$.
Problem 2: Given a graph $G = (V, E)$, find the number of minimal vertex ...
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$\#NAE2SAT$ and $\oplus NAE2SAT$ complexity
Deciding $2SAT$ is in $NL$ and $\#2SAT$ is $\#P$ complete while $\oplus2SAT$ is $\oplus P$ complete.
Deciding $SAT$-$2$-$NAE$ - every clause has exactly $2$ literals, is there an $NAE$ satisfying ...
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Complexity of projected model counting
Counting the models of propositional quantifier-free formulas, #SAT, is #P-complete (under parsimonious reductions).
I was wondering about the complexity of counting the models of existentially ...
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What is known about $\mathrm{NP}^{\mathrm{PP}[1]}$?
Following early discussion on Complexity of maximizing the number of models in a parametric formula it seems that the problem discussed is equivalent to a complete problem in $\mathrm{NP}^{\mathrm{PP}[...
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Complexity of maximizing the number of models in a parametric formula
Let $F(x,y)$ be a propositional formula where $x$ and $y$ are vectors of Booleans. We want to maximize over $x$ the number of models of $F$ over $y$. As a decision problem, this becomes: given $F(x,y)$...
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Hashing-based vs almost uniform sampling-based approximate counting
Corollary 3.6 in the UniqueSAT paper by Valiant and Vazirani [1] states:
For any $\varepsilon > 0$ there is a randomized polynomial-time TM with a SAT oracle, which given a SAT formula $f$ outputs ...
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Fastest Known Algorithm to Count Acyclic Orientations in a Graph
Given an undirected graph $G$, an acyclic orientation of $G$ is choice of orientation for each edge of $G$ (turning each edge into an arc) such that the resulting directed graph has no directed cycles....
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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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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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Complexity of counting 3-colourings of planar bounded degree graphs
The following are known:
It is #P-complete to count the number of 3-colourings of a planar graph (with respect to randomized reductions) [1].
For all $d\geq 3$, it is #P-complete to count the number ...
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The complexity of tensor formula evaluation problem over an infinite field
In the paper The complexity of tensor calculus by C. Damm, M. Holzer and P. McKenzie (link), the authors reduced the problem of computing the permanent of 0/1-matrices to that of evaluating tensor ...
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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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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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$CH=UL$ 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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Are there analogous works to PPSZ algorithm for #P?
The PPSZ algorithm tells us that we can do SAT-solving for
$k-$CNF in time at-most $2^{1-(1-o(1))\frac{\pi^2}{6k}}$.
My question is that do we know such results for counting problems in class #P too ? ...
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Knowing if there are two solutions to the subset sum problem
I was wondering if there are any results that say how hard it is to answer the question are there TWO subsets that sum to a fixed value? In other words, the subset sum problem but asking if there are ...
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Counting subsets of bipartite graph part which admit an induced perfect matching
Given a bipartite graph $G=(U \sqcup V, E)$, count $U^\prime \subseteq U$ for which $\exists V^\prime \subseteq V$ such that the induced subgraph $G[U^\prime \sqcup V^\prime]$ contains a perfect ...
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Approximate counting with non-uniform sampler
Suppose we have a predicate $\phi:\{0,1\}^n\to \{0, 1\}$ corresponding to a self-reducible problem and a FPAUS that can approximately sample from a distribution $p$ such that $\|p-u\|_{L_1}\leq \delta$...
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How does complexity of a counting problem influence wether it admits a closed form formula or not?
In https://arxiv.org/abs/1412.1505, the section "Results on Data Complexity" mentions the fact that since the authors are about to proove $\#P_1$ complexity for weighted model counting in ...
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Is there an oracle that separates $PH$ from counting classes?
Is there an oracle $A$ for which $P^A =PH^A \neq CH^A = NEXP^A$ holds?
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Why is the reduction from 3-SAT to 3-dimensional Matching Parsimonious?
In this talk at the Simons Institute, Holger Dell notes that there is a parsimonious reduction from 3-SAT to the 3-dimensional Matching (3-DM) problem. In other words, there is a reduction between ...
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Consequences of turning $\oplus \text{SAT}$ into few satisfying assignments
Suppose there is a reduction which, given a $\oplus \text{SAT}$ instance $\phi$, returns another $\oplus \text{SAT}$ instance $\psi$ having all the following properties:
The size of $\psi$ is ...
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On solving Planar Circuit SAT
This enquiry is three-sided.
Side 1 - State of the art
Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$?
Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$ assuming ...
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Disproving $\oplus$ETH by reducing $\oplus k$-SAT with $n$ variables and $m$ clauses to planar graph with $o(m^2)$ vertices?
In this question and its answer, they discuss about reducing CNF-SAT with $n$ variables and $m$ clauses to a (problem on) planar graph $G=(V,E)$ with $|V|$ as small as possible. It is said that the ...
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Counting on grid graphs
Are there problems defined on graphs, such as counting 2-factors, Hamiltonian cycles, connected spanning subgraphs etc., that are in $\#P$ and remain hard for grid graphs?
Since there seem to be ...
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The Edge Cover Equilibrium Problem
Let the Edge Cover Equilibrium Problem be the following:
INPUT: a simple undirected graph $G$.
OUTPUT:
YES, if the number of edge covers of $G$ having odd cardinality is equal to the number of edge ...
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Complexity of Model Enumeration in function free, equality free, First Order Logic with only Unary Predicates?
This question has inspired the following two questions.
Given a first order logic language, with only unary predicates, finite number of variables, $\forall$ and $\exists$ i.e no equality and ...
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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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Constructing FOL formula for which counting is easy?
Given a function free First Order Logic language $\mathcal{L}$ are there ways to write formulas for which counting the number of models for a given cardinality of the domain is easy (preferably exists ...
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does within the "range a and b" include a and b?
I have not found the answer to this doubt of mine elsewhere, hence posting it here.
It may be a silly question but I just want to be sure :P
would be great if someone could help me out with this ...
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Asymptotic Approximation to Number of Knapsack Solutions
Is there an asymptotic approximation to the fraction of sets satisfying a knapsack feasibility constraint?
More precisely, imagine I have a large number $n$ of items with bounded weights $X_1,...,X_n ...
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