# Questions tagged [counting-complexity]

How hard is counting the number of solutions?

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### Polylog-space vs NP

Let $\text{polyL} = \cup_{c} \text{SPACE}[\log^c n]$ be the set of all problems that can be solved using polylog space, what is known/believed about its relation with NP? And perhaps even PP? I'm ...
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### $P^{\#P}$ complete problems

Are there any known $P^{\#P}$ complete problems? The problem would have to be at least as hard as anything in the polynomial hierarchy, but perhaps not as hard as PSPACE
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### Working out the constants and probabilities of Stockmeyer's approximate counting algorithm

Stockmeyer's 1983 result on approximate counting using a randomness states that if we have some SAT instance $x$ with $C(x)$ satisfying assignments, then we can find the minimum set of $m$ hash ...
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### Proving #P-hardness for the number of subsets of a set of positive integers with a sum of at most T?

Consider the given problem: you have a set S of positive integers, and you want to find how many subsets have a sum of at most T. I highly suspect that the problem is hard since a polynomial time ...
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### What is the fastest algorithm for computing exact network reliability?

In the network reliability problem, we are given an undirected graph $G$ on $n$ vertices and a parameter $p\in (0,1)$, and are tasked with determining the probability that $G$ becomes disconnected (i....
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### Properties of #P functions that a GapP function may violate

I want to show a specific GapP problem is likely not in #P, actually very closely related to this question in terms of the area of mathematics it is from: How can I show a Gap-P problem is outside #P ...
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### Extending Karp Reductions of a Decision Problem to Cook Reductions of the Associated Counting Problem

It seems that most NP-complete decision problems have #P-complete corresponding counting problems, with many examples showing this and no known counterexamples. In Jerrums' lecture notes `Counting, ...
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### Variation of (derandomized) Valiant-Vazirani

I am interested in the following "improvement" of the Valiant-Vazirani reduction. As pointed out here, under the right derandomization assumptions one can obtain a deterministic polynomial-...
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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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### 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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### 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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