Questions tagged [counting-complexity]

How hard is counting the number of solutions?

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63
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1answer
3k views

More on PH in PP?

A recent question by Huck Bennett asking whether the class PH was contained in the class PP, received somewhat contradictory answers (all true, it seems). On one hand, several oracle results were ...
54
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3answers
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Surprising algorithms for counting problems

There are some counting problems which involve counting exponentially many things (relative to the size of the input), and yet have surprising polynomial-time exact, deterministic algorithms. Examples ...
37
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4answers
3k views

Is $PH \subseteq PP$?

We know that the first level of the polynomial hierarchy (i.e. NP and co-NP) is in PP, and that $PP \subseteq PSPACE$. We also know from Toda's Theorem that $PH \subseteq P^{PP}$. Do we know whether $...
31
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2answers
1k views

When does “X is NP-complete” imply “#X is #P-complete”?

Let $X$ denote a (decision) problem in NP and let #$X$ denote its counting version. Under what conditions is it known that "X is NP-complete" $\implies$ "#X is #P-complete"? Of course the existence ...
30
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2answers
999 views

How hard is it to count the number of factors of an integer?

Given an integer $N$ of length $n$ bits, how hard is it to output the number of prime factors (or alternatively number of factors) of $N$? If we knew the prime factorization of $N$, then this would ...
28
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2answers
2k views

How many DFAs accept two given strings?

Fix an integer $n$ and alphabet $\Sigma=\{0,1\}$. Define $DFA(n)$ to be the collection of all finite-state automata on $n$ states with starting state 1. We are considering all DFAs (not just connected,...
27
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0answers
635 views

The complexity of checking whether two DAG have the same number of topological sorts

This problem is highly related to the CNF one. Here is the problem: given two DAG (directed acyclic graphs), if they have the same counting of topological sorts, answer "Yes", otherwise, answer "No". ...
26
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3answers
2k views

Consequences of #P = FP

Which would be the consequences of #P = FP? I'm interested in both practical and theoretical consequences. From a practical point of view, I'm particularly interested in consequences on Artificial ...
26
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3answers
946 views

When is relaxed counting hard?

Suppose we relax the problem of counting proper colorings by counting weighted colorings as follows: every proper coloring gets weight 1 and every improper coloring gets weight $c^v$ where $c$ is some ...
25
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2answers
808 views

Computational complexity of counting induced subgraphs which admit perfect matchings

Given an undirected and unweighted graph $G=(V,E)$ and an even integer $k$, what is the computational complexity of counting sets of vertices $S\subseteq V$ such that $|S|=k$ and the subgraph of $G$ ...
24
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2answers
848 views

Is there a direct/natural reduction to count non-bipartite perfect matchings using the permanent?

Counting the number of perfect matchings in a bipartite graph is immediately reducible to computing the permanent. Since finding a perfect matching in a non-bipartite graph is in NP, there exists ...
24
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0answers
717 views

What are consequences of the collapse of CH?

I don't grasp the full complexity of the counting hierarchy $CH$. I understand $CH$ is in $PSPACE$, and contains $PH$ within its second level, due to the Toda's theorem. But, what would be important ...
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0answers
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Counting Isomorphism Types of Graphs

Polya's counting theorem leads to an algorithm for counting (precisely) the number of isomorphism types of graphs with $n$ vertices in $\exp (\sqrt n )$ steps. From Polya theorem you get a formula ...
23
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4answers
1k views

Survey on #P and/or counting problems

Can anyone suggest a good and recent survey on counting problems and/or problems that are #P.
20
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4answers
850 views

Examples of hardness phase transitions

Suppose we have a problem parameterized by a real-valued parameter p which is "easy" to solve when $p=p_0$ and "hard" when $p=p_1$ for some values $p_0$, $p_1$. One example is counting spin ...
20
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5answers
728 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 $2$-...
20
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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 ...
19
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3answers
9k views

Counting the Number of Simple Paths in Undirected Graph

How can I go about determining the number of unique simple paths within an undirected graph? Either for a certain length, or a range of acceptable lengths. Recall that a simple path is a path with no ...
19
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1answer
824 views

Count the number of spanning trees fast

Let $t(G)$ denote the number of spanning trees in a graph $G$ with $n$ vertices. There is an algorithm that computes $t(G)$ in $O(n^3)$ arithmetic operations. This algorithm is to compute $\frac{1}{n^...
19
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1answer
653 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 ...
18
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1answer
483 views

What is the counting complexity of random 2-SAT?

Has any work been done on how the complexity of random instances of #2-SAT varies with the clause density? That is: how does the difficulty of counting satisfying solutions to a randomly generated ...
17
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1answer
2k views

What are the #P-complete subfamilies of #2-SAT?

Short version. The original proof that #2-SAT is #P-complete shows, in fact, that those instances of #2-SAT which are both monotone (not involving the negations of any variables) and bipartite (the ...
17
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2answers
2k views

Linear diophantine equation in non-negative integers

There's only very little information I can find on the NP-complete problem of solving linear diophantine equation in non-negative integers. That is to say, is there a solution in non-negative $x_1,x_2,...
17
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1answer
498 views

Approximation for counting the number of simple $s$-$t$ paths in a general graph

I have been told that there are some good polynomial time algorithms for approximating the number of simple paths in an directed graph from given starting vertex $s$ to given ending vertex $t$. Does ...
16
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1answer
1k views

The complexity of counting simple paths in a directed graph

Let $G$ be a digraph (not necessarily a DAG) and let $s,t \in V(G)$. What is the complexity of counting the number of simple $s-t$ paths in $G$. I would expect the problem to be #${\mathsf P}$-...
16
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2answers
791 views

Complexity of counting the number of edge covers of a graph

An edge cover is a subset of edges of a graph such that every vertex of the graph is adjacent to at least one edge of the cover. The following two papers say that counting edge covers is #P-complete: ...
15
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3answers
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Counting the number of Hamiltonian cycles in cubic Hamiltonian graphs?

It is $NP$-hard to find a constant factor approximation of longest cycle in cubic Hamiltonian graphs. Cubic Hamiltonian graphs have at least two Hamiltonian cycles. What are the best known upper ...
15
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1answer
438 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 ...
15
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1answer
442 views

Graph decompositions for combining “local” functions of vertex labelings

Suppose we want to find $$\sum_x \prod_{ij \in E} f(x_i,x_j)$$ or $$\max_x \prod_{ij \in E} f(x_i,x_j)$$ Where max or sum is taken over all labelings of $V$, product is taken over all edges $E$ for a ...
15
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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 ...
14
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4answers
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Counting the number of vertex covers: when is it hard?

Consider the #P-complete problem of counting the number of vertex covers of a given graph $G = (V, E)$. I'd like to know if there is any result showing how the hardness of such problem varies with ...
14
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2answers
704 views

Lower bounds on #SAT?

The problem #SAT is the canonical #P-complete problem. It's a function problem rather than a decision problem. It asks, given a boolean formula $F$ in propositional logic, how many satisfying ...
14
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2answers
855 views

#P-complete problem whose decision version is in P

1) Is it possible to have a parsimonious reduction from a #P-complete problem #A to a counting problem #B when (the decision version) A is NP-complete and the B is in P? For example, can there be a ...
14
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3answers
636 views

The complexity of checking whether two CNF have the same number of solutions

Given two CNF, if they have the same number of assignments to make them true, answer "Yes", otherwise answer "No". It is easy to see it is in $P^{\#P}$, since if we know the exact numbers of ...
14
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2answers
277 views

Above #P and counting search problems

I was reading the wikipedia article about the eight queens problem. It is stated that, there is no known formula for the exact number of solutions. After some searching, I found a paper named "On the ...
14
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2answers
534 views

What's the complexity of Median-SAT?

Let $\varphi$ be a CNF formula with $n$ variables and $m$ clauses. Let $t \in \{ 0,1 \}^n$ represent a variable assignment and $f_{\varphi}(t) \in \{ 0, \ldots , m \}$ count the number of clauses ...
14
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3answers
723 views

How can I show a Gap-P problem is outside #P

There are a number of problems in combinatorial representation theory and algebraic geometry for which no positive formula is known. There are several examples I am thinking of, but let me take ...
14
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2answers
411 views

A question to the #P-complete proof of the permanent from Ben-Dor/Halevi

In the paper of Ben-Dor/Halevi [1] it is given another proof that the permanent is $\#P$-complete. In the later part of the paper, they show the reduction chain \begin{equation} \text{IntPerm} \propto ...
14
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1answer
483 views

Log-space reduction from Parity-L to CNOT circuits?

Question. In their paper Improved simulation of stabilizer circuits, Aaronson and Gottesman claim that simulating a CNOT circuit is ⊕L-complete (under logspace reductions). It is clear that it ...
13
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4answers
749 views

Regarding Pfaffian Methods in Counting and Combinatorics

Recently, I was going over an introduction to Holographic Algorithms. I came across some combinatorial objects called Pfaffians. I do not really know much about those at the moment and came across ...
13
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6answers
591 views

Any algorithmic problem has a time complexity dominated by counting?

What I refer to as counting is the problem that consists in finding the number of solutions to a function. More precisely, given a function $f:N\to \{0,1\}$ (not necessarily black-box), approximate $\#...
13
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2answers
830 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, ...
13
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1answer
656 views

Parity-L vs. NL

Parity-L, also known as $\oplus$L, is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths. A ...
13
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2answers
936 views

Counting solutions of Monotone-2CNF formulas

A Monotone-2CNF formula is a CNF formula where each clause is composed by exactly 2 positive literals. Now, I have a Monotone-2CNF formula $F$. Let $S$ be the set of $F$'s satisfying assignments. I ...
13
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1answer
571 views

Is counting maximal cliques in an incomparability graph #P-complete?

This question is motivated by a MathOverflow question by Peng Zhang. Valiant showed that counting maximal cliques in a general graph is #P-complete, but what if we restrict to incomparability graphs (...
12
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2answers
2k views

Complexity of counting all connected subgraphs

Let G be a connected graph. What is the complexity of counting all connected subgraphs if G is of the following types? G is general. G is planar. G is bipartite. I don't care about ...
12
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3answers
1k views

Complexity of counting paths in a graph

Given a directed graph with n nodes such that each vertex has exactly two outgoing edges, and a natural number N encoded in binary, two vertices s and t, I want to count the number of (not ...
12
votes
2answers
1k views

Is $coNP^{\#P}=NP^{\#P}=P^{\#P}$?

By http://www.cs.umd.edu/~jkatz/complexity/relativization.pdf If $A$ is a PSPACE-complete language, $P^{A}=NP^{A}$. If $B$ is a deterministic polynomial-time oracle, $P^{B}\ne NP^{B}$ (assuming $P\...
12
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1answer
888 views

Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$

I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER ...
12
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0answers
137 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, ...

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