Questions tagged [counting-complexity]

How hard is counting the number of solutions?

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9
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0answers
135 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 ...
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101 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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112 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 $g : \{0,1\}^* \to \mathbb{N}$,...
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40 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
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1answer
144 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
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1answer
163 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
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1answer
128 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
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93 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
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1answer
424 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
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1answer
184 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
321 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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77 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
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3answers
289 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
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1answer
578 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
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1answer
113 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 = \...
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60 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
99 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
72 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
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2answers
131 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
176 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 ...
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2answers
154 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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67 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
119 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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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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106 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
149 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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59 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
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1answer
57 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
110 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 ...
5
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193 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
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1answer
339 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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168 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
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1answer
149 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
128 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
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1answer
234 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
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1answer
193 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
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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 ...
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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 ...
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82 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
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1answer
186 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
104 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
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1answer
181 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
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1answer
176 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
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2answers
735 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
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0answers
162 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 ...
3
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1answer
193 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 ...
4
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1answer
171 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. #...
8
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1answer
194 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
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1answer
252 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-...
4
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1answer
115 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 ...