# Questions tagged [counting-complexity]

How hard is counting the number of solutions?

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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  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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### Phase transition in counting feasible solutions to knapsack problems?

Imagine that you have a normalized knapsack constraint with $n$ items and weights $w_1,...,w_n$ satisfying $\sum_{i=1}^n w_i = 1$. I'm trying to understand the behavior of the function Z(c) = \#| S ...
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### Uniformly sampling or counting connected graph partitions with any number of blocks

Question: Is it possible to uniformly sample in polynomial time from the set of all connected partitions of a graph? Or is there a JVV type argument that proves this to be NP-hard? To clarify: By a ...
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### Average case hardness of #SAT

Is there anything known about the average case hardness of #SAT? Let’s say over a uniform distribution. We know that in the worst case, it is #P-complete, but what can we say about an average ...
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### Is the counting version of 1-in-3 Sat #P-complete?

In the paper "Hard Tiling Problems with Simple Tiles", Moore and Robson prove that Cubic Planar Positive 1-in-3 Sat in NP-complete by a reduction from Positive 1-in-3 Sat. Cubic Planar Positive 1-in-...
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### Is this a known problem, and is it #P-complete?

Let $G=(V,E)$ be an undirected graph. What I call a selection function of $G$ is a partial function $f:V \to E$ such that for every node $v$, if $f(v)$ is defined then it is one of the adjacent edges ...
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### Is counting the total number of faces of a polytope $\#P$ hard?

Let $P$ be a polytope defined by $Ax = b, x \geq 0$. Question: What is the complexity of computing the total number of faces of $P$? I know counting vertices is $\# P$-complete, but this problem is ...
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### Network Reliability Problem

Network reliability, in which we are given an undirected graph $G$ with a failure probability $p_e$ for each edge and we are asked to calculate the probability that the network becomes disconnected ...
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### Complete problems for FP

Let FP be the class of functions $f : \{0,1\}^* \to \mathbb{N}$ that can be computed in polynomial time. Moreover, given two functions $f : \{0,1\}^* \to \mathbb{N}$ and $g : \{0,1\}^* \to \mathbb{N}$,...
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### Updating set of lists dependent upon a few indices

I'm curious about a data structure for a set of "valid lists", where you have a set of lists of length $i$ $S_i$, have a list $L$ of possible items to append, and a boolean function $f$, and wish to ...
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