Questions tagged [counting-complexity]
How hard is counting the number of solutions?
217
questions
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1answer
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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 ...
1
vote
2answers
122 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 ...
3
votes
1answer
82 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 ...
3
votes
0answers
64 views
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 ...
1
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0answers
103 views
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 ...
0
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1answer
98 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-...
5
votes
1answer
184 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 ...
10
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0answers
142 views
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 ...
0
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0answers
104 views
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 ...
3
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0answers
117 views
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}$,...
0
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0answers
42 views
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 ...
5
votes
1answer
157 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 $...
0
votes
1answer
168 views
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 ...
1
vote
1answer
155 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?...
5
votes
0answers
101 views
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 ...
20
votes
1answer
452 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 ...
7
votes
1answer
192 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 ...
4
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2answers
329 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 ...
0
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0answers
79 views
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 ...
4
votes
3answers
329 views
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 ...
15
votes
1answer
654 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 ...
3
votes
1answer
120 views
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 = \...
1
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82 views
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....
3
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1answer
112 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 ...
3
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0answers
76 views
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 ...
5
votes
2answers
143 views
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 $\...
4
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1answer
208 views
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 ...
2
votes
2answers
169 views
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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0answers
69 views
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 ...
5
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1answer
124 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 ...
2
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0answers
29 views
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?
5
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0answers
132 views
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 ...
5
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1answer
153 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 ...
1
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0answers
71 views
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. ...
0
votes
1answer
62 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 ...
-1
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1answer
114 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 ...
6
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0answers
208 views
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 ...
4
votes
1answer
554 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 ...
0
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0answers
170 views
On PP in communication complexity
Aho says $D(f)=O(N(f)N(\overline f))$ where $D(f)$ is deterministic communication complexity and $N(f)$ is non-deterministic version.
Do we know $PP(f)=\Omega(2^{(N(f)N(\overline f))^{O(1)}})$ or $...
2
votes
1answer
162 views
$⊕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%...
0
votes
1answer
134 views
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 ...
9
votes
1answer
251 views
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 ...
7
votes
1answer
203 views
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 ...
3
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0answers
113 views
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 ...
3
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0answers
113 views
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 ...
0
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0answers
83 views
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 $\Bbb F_{3^t}$ ...
2
votes
1answer
198 views
What does $\#P\subseteq FP^{PPAD}$ imply?
We know $\#P\subseteq {PPAD}\implies PH\subseteq P^{{PPAD}}\subseteq P^{{NP}}$ and the polynomial hierarchy collapses ($FP^{PPAD}=PPAD$ following Emil Jerabek's comment).
Can $\#P\subseteq {PPAD}$ ...
0
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1answer
133 views
What is known about counting bipartite perfect matching with average degree in $[2,3]$ and max degree $3$?
We know counting perfect matching for bipartite graphs with vertex degree $2$ is in $P$ while counting perfect matching for graphs with vertex degree $3$ is in $\#P$.
We also know there are degree $3$...
7
votes
1answer
188 views
Composition of $FP$ and $\#P$ functions
Let $f_i \in FP$ and $g_i \in \#P$ for $i \in \mathbb{N}$.
It is known that: $f_1(f_2(x)) \in FP$ and that $g_1(f_1(x)) \in \#P$.
Is it known whether or not $f_1(g_1(x)) \in \#P$ or maybe $f_1(g_1(...
-1
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1answer
196 views
#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 ...
