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How hard is counting the number of solutions?

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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
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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
137 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
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 ...
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1answer
102 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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1answer
192 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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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 $...
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An oracle that separates BPP from PP?

Is there an oracle that gives $P=BPP\neq NP=PP$?
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46 views

Does Ladner's theorem work in $\oplus P$ and $PP$ setting?

The answer here (Generalized Ladner's Theorem) says for many natural settings Ladner's theorem applies. For many interesting problems such as those equivalent to finding parity of number of ...
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1answer
132 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%...
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1answer
89 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 ...
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1answer
211 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 ...
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1answer
179 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 ...
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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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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 $\Bbb F_{3^t}$ ...
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1answer
170 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}...
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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 $3$...
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164 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(...
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1answer
118 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 ...
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581 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, ...
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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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1answer
164 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 ...
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1answer
129 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. #...
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1answer
181 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.
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164 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-...
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1answer
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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 ...
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2answers
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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 ...
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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 $\...
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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 $2$-...
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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 ...
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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{...
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Efficient algorithms for counting $k$-clique subgraphs

Given a graph $G$ with $n >> k$ vertices, what are the fastest algorithms known to count the number of induced subgraphs in $G$ that are $k$-cliques? Are there algorithms that can do better ...
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1answer
160 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 ...
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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 ...
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#P-completeness and ModkP-completeness

If $\#A$ is the counting version of some corresponding decision problem $A$, when, if ever, can we determine solely on the basis of the complexity of the underlying decision problem that $\#A$ is #P-...
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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 ...
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1answer
606 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 ...
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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? $\...
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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, ...
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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 \...
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239 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 ...
12
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1answer
303 views

Sampling a uniformly random satisfying assignment

Problem: Given $\phi : \{0,1\}^n \to \{0,1\}$ represented by a boolean circuit, generate a uniformly random $x \in \{0,1\}^n$ such that $\phi(x)=1$ (or output $\perp$ if no such $x$ exists). Clearly ...
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Consequences of VP = VNP on randomness

According to the answers in posting it is possible that $\mathsf{VP} = \mathsf{VNP}$ and $\mathsf{P} \neq \mathsf{NP}$ are simultaneously correct. $\mathsf{VP} = \mathsf{VNP}$ implies $\mathsf{P/...
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What is largest class of functions $C$ such that we know $\#P$ in not contained in $C$-generated $TC^0$?

In [1] it is stated that "It remains an open question as to whether every function in $\#P$ has $TC^0$ circuits (although it is at least known that not all $\#P$ functions have DLogTime-uniform $...
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Can weakly parsimonious counting reductions use the input instance to compute the count?

Quick version: is there a definitive definition of weakly parsimonious counting reduction for #P? Longer version: I am doing a gadget based reduction for NPC and would like to use it for #P. The ...