Questions tagged [counting-complexity]

How hard is counting the number of solutions?

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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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75 views

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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127 views

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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85 views

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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1answer
110 views

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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191 views

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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158 views

Counting avoiding improper 3-colorings

Given a graph $G=(V,E)$, what I call an improper $3$-coloring of $G$ is simply a function $c:V \to \{1,2,3\}$. I say that $c$ is $1$-$2$-avoiding when there do not exist two adjacent nodes $u,v$ with $...
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Language in $PSPACE$ and not necessarily in $P$ if $P=PP$?

If $P=PP$ then the counting hierarchy collapses to $CH=P$. Because so many complexity classes are contained in $CH$, this causes most classes to now be contained in $P$. My question is whether this is ...
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161 views

Count satisfying assignments of CNF formulas over all possible negation assignments

Consider the set of all CNF instances that can be generated by adding negations to a single monotone CNF instance. How hard is it to compute the sum of the counts of satisfying assignments for the set?...
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Counting matchings on 3-regular bipartite graphs

What I call a graph here allows parallel edges. Is the following problem #P-hard: INPUT: a 3-regular bipartite graph $G$ OUTPUT: the number of matchings of $G$. It is known that counting matchings ...
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455 views

Is prime-counting function #P-complete?

Recall $\pi(n)$ the number of primes $\le n$ is the prime-counting function. By "PRIMES in P", computing $\pi(n)$ is in #P. Is the problem #P-complete? Or, perhaps, there is a complexity reason to ...
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199 views

Holant problems and holographic reduction: simple graphs or multigraphs?

From what I can understand, Holographic reductions for Holant problems are used to show #P-hardness or polynomial time computability of certain counting problems on undirected graphs that have very ...
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333 views

A variant of #POSITIVE-2-DNF

Let $G=(V,E)$ be an undirected graph. I call a valuation of $G$ a function $\nu: V \to E$ that maps every node $x \in V$ to an edge incident to $x$ (so that there are $\prod_{x \in V} d(x)$ valuations ...
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Given $n\times n$ matrix $A$ with integer entries, find #$k$SAT formula that yields $\mathrm{perm}(A)>0$

For each #$k$SAT instance one can build a matrix $A$ such that $\mathrm{perm}(A) = F(\Sigma)$, where $\Sigma$ is the solution count of the $k$SAT formula and $F$ an easy to invert function. My ...
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Is counting simple cycles in $P$ for graphs of bounded tree width?

Motivation: Determining if a graph has a Hamiltonian cycle is $NP$-hard in general. However, determining if there is a Hamiltonian cycle is in polynomial time on graphs of bounded tree width, either ...
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663 views

Can one efficiently uniformly sample a neighbor of a vertex in the graph of a polytope?

I have a polytope $P$ defined by $\{ x : Ax \leq b, x \geq 0\}$ . Question: Given a vertex $v$ of $P$, is there a polynomial time algorithm to uniformly sample from the neighbors of $v$ in the graph ...
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Complexity of #PP2DNF where we also count on the number of clauses

The #PP2DNF problem is the following: we have variables $X = \{x_1, \ldots, x_n\}$, $Y = \{y_1, \ldots, y_n\}$, and a positive partitioned 2-DNF formula, i.e., a Boolean formula of the form $\phi = \...
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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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1answer
118 views

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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Concrete examples of $\sharp P_1$ complete problems? Self avoiding walks?

The only examples of $\sharp P_1$ complete problems I've seen are fairly abstract : e.g. here https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/enumerate.pdf Valiant proves that there exists a $\...
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Why is counting the number of hamiltonian subgraphs $\sharp P $ hard?

I'm confused about how to prove either of the following closely related statements. They are both from this paper: https://epubs.siam.org/doi/10.1137/0208032 1) "A further problem that can be shown ...
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Is S-T CONNECTEDNESS #P-complete on instances when all s-t paths are of the same length?

S-T CONNECTEDNESS Input: a (undirected) graph $G=(V,E)$; $s,t \in V.$ Output: number of spanning subgraphs of $G$ in which there is a path from $s$ to $t$. S-T CONNECTEDNESS problem is known to be #...
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Complexity class of approximating perfect match count

We know we can approximate perfect matching count of bipartite and approximate volume of convex bodies in randomized polynomial time. Is there any evidence these approximations could be in Nick's ...
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1answer
128 views

Counting/Enumerating Minimal Edge Covers

A Minimal Edge Cover is an Edge Cover such that no other Edge Cover is a proper subset of it. Questions Which is the complexity of counting Minimal Edge Covers? Do we know any non-trivial ...
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Heuristics for exact #3COLORING close to the 3-colorability threshold

What are some fast heuristics for exactly counting 3-colorings of graphs close to or at the 3-colorability threshold? Is there literature on the average-case performance for any of these methods?
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Parsimonious Reduction from Unique-3SAT to NAE-3SAT

Using the result by Valiant and Vazirani, we know that Unique-3SAT (3SAT with a unique solution) is hard unless NP=RP. Also it is widely believed that the "Unique" version of any NP-complete problem ...
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1answer
154 views

Sets of solutions which it is hard to uniformly sample from, but easy to integrate functions over? (Or compute expectations over?)

I'm curious if there is a problem (e.g. something like perfect matchings on a given graph, number of solutions to a boolean equation, etc. for precise frameowork see JVV86) such that: 1) It is hard ...
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Counting vertex covers on a chain of k nodes that do not contain a sub-chain of length >=3

By a "chain of k nodes", I mean k nodes lined up like a linked-list: o-o-o.....-o . By "do not contain a sub-chain of length >=3", I mean that no cover should contain two edges that shares a node. ...
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1answer
64 views

What is the deterministic complexity of counting the number of global minimum cuts on an unweighted undirected graph?

I know as a consequence of Karger's algorithm that the number of minimum cuts is bounded by $\binom{n}{2}$. In the comments of Counting the number of distinct s-t cuts in a oriented graph It says ...
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115 views

Special cases of hard counting problems that are easy

We know that bipartite planar perfect matching count is easy, permanent mod $3^t$ is easy for orthogonal matrices, permanent mod $2$ is easy, bounded rank permanent is easy. Outside of permanent ...
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How hard is APPROXIMATE-#SAT?

It is well known that the problem of counting the satisfying assignments of SAT, namely the problem #SAT, is #P-complete. It is also suspected (somewhat less widely) that even deciding SAT should ...
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741 views

What is the complexity of counting parse trees?

A Counting Problem Given a CFG $G$ and a string $s$, how many distinct parse trees are there for the string $s$? An Example Instance Let's consider an example instance consisting of a CFG $G$ with ...

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