# Questions tagged [counting-complexity]

How hard is counting the number of solutions?

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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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### Problems rephrased as quadratic unconstrained binary optimization

I was impressed when i came across Quadratic unconstrained binary optimization (QUBO) recently, and saw how one can rephrase many combinatorial problems into questions about optima of binary functions....
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### Computational hardness for sampling a uniform matching

A famous result of Jerrum, Sinclair, and Vigoda shows that there exists a polynomial-time algorithm which takes a bipartite graph $G$ and produces a random perfect matching $M$ of $G$ (assuming one ...
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### Is #PP2DNF hard to approximate?

The problem #PP2DNF asks to count the number of satisfying assignments of a positive partitioned 2-DNF Boolean formula, i.e., a formula $\phi$ on variables $X_1, \ldots, X_n, Y_1, \ldots, Y_m$ of the ...
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### $⊕P$-completeness of$⊕2SAT$

Is $⊕2SAT$ - the parity of the number of solutions of $2$-$CNF$ formulae $\oplus P$ complete? This is listed as an open problem in Valiant's 2005 paper https://link.springer.com/content/pdf/10.1007%...
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### On polytope lattice points

Given a convex polytope let the width of the polytope be $d$ and the farthest euclidean distance between any points in the polytope be $e$. Denote $\mathcal P(a,c)$ to be the set of convex polytopes ...
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### Does $NP=PP$ collapse the counting hierarchy?

Suppose $NP=PP$. Then a simple argument shows that $PH^{PP}=NP$. Can we go one step further and get $PP^{PP}=NP$? The simple argument is Theorem If $NP=PP$ then $PH^{PP}=NP$. Proof $PP$ is closed ...
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### How to benchmark #2-SAT counting algorithms?

Are there any libraries of #2-SAT instances that are hard to solve with state-of-the-art exact solvers (sharpSAT, cachet, ...)? Alternatively: are there practical ways to generate hard #2-SAT ...
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### On approximating problems in $\#P$

We know that for every counting problem $\#A$ in $\#P$, there is a probabilistic algorithm $\mathcal C$ that on input $x$, computes with high probability a value $v$ such that (1 − ε)\#A(x) ≤ v ≤ (1 ...
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### Is Permanent $+$-reducible?

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$. Is ...
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### On permanent mod $3^t$ computation

By $'$ I mean transpose. I gather the info here from rjlipton.wordpress.com/2014/12/21/modulating-the-permanent. We know that if $U\in\Bbb F_{3^t}^{n\times n}$ satisfies $UU'=I_n$ in $\Bbb F_{3^t}$ ...
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### #P-complete with decision P

We know that an NP complete problem with a parsimonious reduction but a many one reduction is candidate problem for NP complete problem not being #P hard. Likewise is there a possible classification ...
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### Counting the number of satisfying assignments in a POSITIVE CNF-SAT

We know the problem of counting the number of satisfying assignment in a given general boolean formula (CNF-SAT), a given DNF formula, or even a given 2SAT formula is a #P-complete problem. Now, ...
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### On space complexity of permanent modulo $2^t$?

We know from here that permanent of $0/1$ matrix modulo $2^t$ is in $DTIME(n^{t+3})$ and hence in $P$. My question is whether permanent of $0/1$ matrix modulo $2^t$ is in $L$ as well or is the current ...
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### Where is the counting hierarchy if polynomial hierarchy collapses?

Supposing if $P^{\#P}\subseteq BPP$ then polynomial hierarchy collapses. Does the counting hierarchy collapse as well? Irrespective of $P^{\#P}\subseteq BPP$ are there any collapse results of ...
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### Is #CYCLE #P-complete?

We know that #SAT is #P-complete. We also know that problems with polynomial decision versions like PERMANENT are #P-complete. Is it true that finding the number of simple cycles in a graph, i.e. #...
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### Was counting complexity first introduced by Valiant in 1979?

Was #P first introduced in ?  Valiant, Leslie G. "The complexity of computing the permanent." Theoretical computer science 8.2 (1979): 189-201.
Consider the following problem: Given is a multiset of positive integers, $S$, and an integer $k$. Count submulisets of $S$ of size $k$, $\{s_1,\dotsc,s_k\} \subseteq S$, such that when the $s_i$ are ...