Questions tagged [counting-complexity]
How hard is counting the number of solutions?
11
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1answer
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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
0answers
47 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
vote
0answers
34 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....
4
votes
1answer
85 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
votes
0answers
65 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
113 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 $\...
2
votes
0answers
100 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
137 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 #...
1
vote
0answers
63 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 ...
6
votes
1answer
108 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
votes
0answers
25 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
votes
0answers
90 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
votes
1answer
143 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
vote
0answers
40 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
51 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
votes
1answer
106 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 ...
4
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0answers
182 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 ...
5
votes
1answer
235 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
votes
0answers
107 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 $...
3
votes
1answer
139 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
100 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 ...
8
votes
1answer
225 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
187 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 ...
2
votes
0answers
107 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
votes
0answers
111 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
votes
0answers
80 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
175 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
votes
1answer
95 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
173 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(...
0
votes
1answer
128 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 ...
13
votes
2answers
668 views
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, ...
5
votes
0answers
153 views
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 ...
4
votes
1answer
177 views
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 ...
5
votes
1answer
141 views
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. #...
9
votes
1answer
187 views
Was counting complexity first introduced by Valiant in 1979?
Was #P first introduced in [1]?
[1] Valiant, Leslie G. "The complexity of computing the permanent."
Theoretical computer science 8.2 (1979): 189-201.
6
votes
1answer
210 views
What's the relationship between ASP-complete and #P-complete?
Given that ASP-reductions by definition are parsimonious and parsimonious reductions preserve #P-completeness, one might think that the counting version of all ASP-complete problems are also #P-...
5
votes
1answer
108 views
Counting subsets of given set with certain properties
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 ...
5
votes
2answers
154 views
Complexity of Uniform Generation of Perfect Matchings
Jerrum,Valiant and Vazirani on their paper "Random generation of combinatorial structures from a uniform" (http://www.cc.gatech.edu/~vazirani/AppCount.pdf) talk about seeing problems related to ...
1
vote
1answer
180 views
Complexity of permanent modulo prime
Given $M\in\Bbb Z^{n\times n}$ with $O(n)$ bit entries (could be all in $\{0,1\}$), $p$ a prime of $O(n^\alpha)$ bits for some $\alpha\in(0,1]$ and a $c,d\in\Bbb Z$ with $0\leq c<d<p$, is 'Is $\...
21
votes
5answers
666 views
Easy problems with hard counting versions
Wikipedia provides examples of problems where the counting version is hard, whereas the decision version is easy. Some of these are counting perfect matchings, counting the number of solutions to $2$-...
10
votes
1answer
387 views
Status of PP-completeness of MAJ3SAT
SHORT QUESTION: Is MAJ-3CNF a PP-complete problem under many-one reductions?
LONGER VERSION:
It is well-known that MAJSAT (deciding whether the majority of assignments of propositional sentence ...
5
votes
0answers
141 views
Is this volume computation problem #P Hard?
Let $A_{n\times n}$ be a positive definite diagonal matrix with positive rational entries, and let $b$ be a positive rational. Let $R(A,b)$ be the ellipsoid
$ \{\mathbf{x}\in \mathbb{R}^n : ||A\mathbf{...
6
votes
1answer
177 views
Complexity of #SAT for monotone DNF formulae whose hypergraph is a hypertree
A monotone DNF on variables $x_1, \ldots, x_n$ is a disjunction of clauses, each clause being a conjunction of some of the $x_1, \ldots, x_n$. The #SAT problem asks, given a monotone DNF $\Phi$, how ...
13
votes
3answers
708 views
Complexity of computing the parity of read-twice opposite CNF formula ($\oplus\text{Rtw-Opp-CNF}$)
In a read-twice opposite CNF formula each variable appears twice, once positive and once negative.
I'm interested in the $\oplus\text{Rtw-Opp-CNF}$ problem, which consists in computing the parity of ...
9
votes
2answers
1k views
Complexity of counting matchings in a bipartite graph
I might be missing something obvious but I can't find references about the complexity of counting matchings (not perfect matchings) in bipartite graphs. Here is the formal problem:
Input: a bipartite ...
19
votes
1answer
615 views
Division by two of functions in #P
Let $F$ be an integer valued function such that $2F$ is in $\#P$. Does it follow that $F$ is in $\#P$? Are there reasons to believe this is unlikely to always hold? Any references I should know ...
3
votes
0answers
166 views
Big picture in counting complexities
(1) Is there a relation ( conjectured relation) between $\mathsf{\#P}$ and $\mathsf{CH}$?
(2) How does $\tau$ conjecture in complexity of factorial fit in the picture?
Is there a good reference?
$\...
12
votes
0answers
120 views
Is it #P-hard to compute the number of antichains of a distributive lattice?
An antichain of a poset $(P, <)$ is a subset of pairwise incomparable elements, namely, a subset $A \subseteq P$ such that there are no $x, y \in A$ with $x < y$. By a result of Provan and Ball, ...
2
votes
0answers
389 views
Complexity of counting spanning subgraphs that have cycle properties
Let $G = (V, E)$ be a connected undirected graph. I am interesting in counting the number of connected spanning subgraphs having:
$\ge 1$ cycle each (but can have any number of cycles),
Exactly $k \...
8
votes
1answer
241 views
What's the complexity of counting odd nodes in graph?
According to Handshaking Lemma: any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number. This observation means that if we are ...
