How hard is counting the number of solutions?

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8
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1answer
152 views

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 ...
4
votes
0answers
35 views

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 ...
3
votes
3answers
195 views

Milestones in counting complexity

I want to prepare a small presentation with some of the most important results in Counting complexity, continue with recent ones and finish with some interdisciplinary results ,probably with other ...
3
votes
1answer
203 views

#P- vs PP-Completeness

Suppose $A$ is any #P-complete problem. Now, $A$ is modified to obtain a decision problem $A'$ not by asking whether there is a solution but whether at least half of the potential solutions are ...
5
votes
1answer
180 views

Easy decision hard counting Parametrized

It is known that counting perfect matchings in a bipartite graph is #P-complete. On the other hand, finding a perfect matching belongs in P. Is there a problem, that exhibits the same behavior in ...
2
votes
0answers
53 views

Hardness of approximately counting independent sets with a PRAS, rather than FPRAS

It is known that approximately counting the independent sets of a graph is hard, even if randomness can be used, and even if we restrict ourselves to bounded degree graphs with degree bound at least ...
20
votes
0answers
485 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 ...
1
vote
1answer
109 views

Almost uniform sampling implies approximate counting

I began studying papers about approximate counting and I keep seeing the above being quoted, without further explanation. I suppose the procedure that yields the result is very well known and that is ...
12
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2answers
168 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 ...
8
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0answers
75 views

Is there a counting complexity class for succint problems?

Encoding NP-complete problems succintly often makes them NEXP-complete. I am wondering if counting the number of solutions to such a problem with a succint encoding would be any harder than solving ...
6
votes
1answer
106 views

FPRAS on #P complete problems and self reducibility

I am quoting a phrase of Martin Dyer in his paper Approximate Counting by Dynamic Programming: Since 0-1 knapsack is self-reducible, existence of an fpras for the problem now follows indirectly from ...
4
votes
1answer
128 views

Deciding whether the sum of independent random variables exceeds a threshold a majority of the time, PP-hard?

Say I have $n$ independent Bernoulli random variables, with parameters $p_1,\ldots,p_n$. Say, also, that I wish to decide whether their sum exceeds some given threshold $t$ with probability at least ...
0
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2answers
203 views

Upper bound for number of independent sets

What is the tightest upper bound known for the number of independent sets in a graph?
1
vote
0answers
119 views

estimating the number of comparisons of Shell Sort

I would like to estimate the number of comparisons in ShellSort. I'm using $h_s = 2^s-1$, where $s=\left \lfloor{\log(n)}\right \rfloor, \left \lfloor{\log(n)}\right \rfloor -1, \dots, 1 $ ; I know ...
13
votes
1answer
566 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 ...
7
votes
0answers
513 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 ...
1
vote
1answer
380 views

#P-complete problems are at least as hard as NP-complete problems

I just read J. Scott Provan, Michael O. Ball: The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM J. Comput. 12(4): 777-788 (1983) and one of the first ...
10
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0answers
211 views

Can we approximate the number of words accepted by an NFA?

Let $M$ be an acyclic NFA. Since $M$ is acyclic, $L(M)$ is finite. In a related question, it was suggested that exact counting of the number of words accepted by $M$ is $\#P$-Complete. The second ...
7
votes
2answers
133 views

Is it known whether counting $q$-dimensional $p$-matching is $\#W[1]$-Hard?

The $q$-Dimensional $p$-Matching is defined as follows: Given disjoint universes $U_1,\ldots,U_q$, think of an element in $U_1\times\ldots\times U_q$ as a set that contains exactly one element from ...
10
votes
1answer
380 views

Probability of generating a desired permutation by random swaps

I'm interested in the following problem. We're given as input a "target permutation" $\sigma\in S_n$, as well as an ordered list of indices $i_1,\ldots,i_m\in [n-1]$. Then, starting with the list ...
4
votes
1answer
387 views

Example of #P-intermediate problem

The previous question Do there exist intermediate problems (in the sense of Ladner's Theorem) for FP vs. #P? I assume that something is known, because I read some papers concerned with FP/#P ...
0
votes
1answer
135 views

Does $\# \mathsf{P}\subseteq \mathsf{FP}^{\mathsf{PH}}$?

The Toda's theorem is a relationship between two different complexity classes: $ \# \mathsf{P} $ and $PH$. He proved that $ \mathsf{PH}\subseteq \mathsf{P}^{\#\mathsf{P}} $. I wonder the following ...
6
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2answers
161 views

Is there an oracle separating Parity-P from PSPACE?

Is $ (\oplus \mathsf{P})^A \not \supseteq \mathsf {PSPACE}^A$ for some language $A$?
9
votes
1answer
146 views

Complexity of counting graph endomorphisms

A homomorphism from a graph $G = (V, E)$ to a graph $G' = (V', E')$ is a mapping $f$ from $V$ to $V'$ such that if $x$ and $y$ are adjacent in $E$ then $f(x)$ and $f(y)$ are adjacent in $E'$. An ...
7
votes
1answer
160 views

Complexity of counting poset automorphisms

A (finite) poset $P = (X, <)$, or partially ordered set, is a (finite) set $X$ equipped with a transitive antisymmetric relation $<$; it can be equivalently seen as a DAG $G = (X, E)$ that is ...
8
votes
2answers
193 views

Approximating #P-hard problems

Consider the classical #P-complete problem #3SAT, i.e., to count the number of valuations to make a 3CNF with $n$ variables satisfiable. I am interested in the additive approximability. Clearly, there ...
0
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0answers
82 views

Is there a useful notion of pathwidth-treewidth for posets?

Consider a poset $P = (V,A)$. We may define a path structuring of $P$ as a chain $\Sigma$ of the form $X_0 \subset X_1 \subset \ldots \subset X_n$ where : (i) for every $x \in V$, the set $\{ i \in ...
3
votes
0answers
181 views

#EXP-Complete problems

Let #EXP be the counting variant of NEXP, in the same way that #P is the counting variant of NP. Are there any known #EXP-complete problems? In particular, has #Succinct Sat (the natural candidate) ...
0
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0answers
43 views

Can there exist a single Turing machine complete for PTIME, or for $\#P_1$?

In "The Complexity of Enumeration and Reliability Problems", Valiant mentions the existence of a single Turing machine that is complete for the class $\#P_1$ (i.e., $\#P$ with unary input). On page ...
3
votes
0answers
134 views

Counting problems with two-polymatroids

Let $S$ be a finite set. Following Lovasz (Matroid matching and some applications), let us define a polymatroid function over $S$ as a function $f : 2^S \rightarrow \mathbb{N}$ such that (1) ...
10
votes
1answer
175 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: ...
5
votes
1answer
197 views

Solve multiple instances of SAT with a 2-approximating #SAT query

Assume we have some oracle $A$ such that when given as input a Boolean formula $\phi$, it outputs a 2-approximation to the number of satisfying assignments of $\phi$. If we are given multiple SAT ...
4
votes
0answers
88 views

Variant of Toda's theorem for intermediate levels of the polynomial hierarchy

Is there a version of Toda's theorem for intermediate levels of the polynomial hierarchy ? More precisely, is there any variant of the Toda's theorem that states: Let $\# wSAT$ be the number of ...
3
votes
0answers
95 views

Gradual increase in hardness from P to PH of #SAT

We know that counting the number of solutions to $3-SAT$ is the canonical $\#P-Complete$ problem. Equivalently, it is the canonical $PH-Hard$ problem. However, counting the number of solutions to ...
2
votes
0answers
60 views

Weight enumerator and levels of polynomial hierarchy

Let $A_i$ be the number of codewords in a binary linear code $\mathcal{C}$ of weight $i$. It is known that: $A_k$ is in $P$, where $k = \mathcal{O}(\log_2 n)$. $A_{n}$ is in $\#P-Complete$, ...
3
votes
0answers
112 views

Deciding whether a binary multiplicity automaton has empty language

Multiplicity automatons (see here) is an interesting model. They have the (almost) same syntax as a non-deterministic finite automatons, but instead of deciding whether a word belongs to a language, ...
3
votes
0answers
104 views

Computing the permanent with polylog size matrices

The complexity of computing the permanent of a $l\times l$ binary matrix is known to be $\#\mathsf{P}$-complete, from the famous result of Valiant, where $l = \Theta(n)$. We know that the problem is ...
13
votes
3answers
4k 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 ...
5
votes
1answer
359 views

Do we know whether P^#P = NP^#P?

I thought the relation between P using a #P-oracle and NP using a #P-oracle is still unknown (or equivalently the relation between P^PP and NP^PP). Recently, I have read in a journal article that ...
5
votes
0answers
112 views

Relativization of Toda's Theorem

I'm trying to figure out some consequences of the fact that Toda's Theorem relativizes. The (un-relativized) Toda's theorem states that $PH \subset P^{\#P}$ so that for any constant $k$ and any ...
4
votes
0answers
160 views

The weakly NP-complete problems and their associated counting problem

Are there weakly NP-complete problems whose associated counting problem can be computed in pseudo-polynomial time? And if one were to be found (and assuming it is #P-complete), what would be the ...
16
votes
1answer
1k 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 ...
0
votes
1answer
54 views

Lower bounds on counting functions

I have a question about counting problems on arbitrary (not necessarily polynomial time) functions. Let $F_n = \{f : \{0,1\}^n \to \{0,1\}\}$ be the set of all boolean functions with $n$ inputs ...
5
votes
0answers
226 views

Weight enumerator of a binary linear code

The weight enumerator polynomial of a $(n,k)$ binary linear code $\mathcal{C}$ is defined as $$WE(\mathcal{C}) = \sum_{i=0}^{n}WE_{i}(\mathcal{C}) x^{i}$$ where $$WE_{i}(\mathcal{C}) = ...
10
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2answers
243 views

How hard is it to count the number of local optima for a problem in PLS?

For a polynomial local search problem, we know that at least one solution (local optimum) must exist. However, many more solutions could exist, how hard is it to count the number of solutions for a ...
17
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0answers
431 views

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 ...
3
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0answers
166 views

What NP-complete problems are expected not to have #P-hard counting problems? [duplicate]

Let $R(v_{\bullet}, w_{\bullet})$ be some $P$-time computable relation between two binary strings $v_{\bullet}$ and $w_{\bullet}$. $NP$ problems are problems of the form: Given $v_{\bullet}$, ...
3
votes
1answer
131 views

Complexity class for Optimization problems over #P functions

Is there any complexity class which contains problems that can be expressed as an optimization over polynomially many #P functions ? i.e: $$\tilde{f}(x) = \text{Max}_{f \in F}f(x)$$ where $f\in\# ...
7
votes
2answers
367 views

What are the current best upper bounds of #P?

#P is the class of counting problems for problems in NP. In other words, a solution to #P returns the number of solutions to a particular problem in NP. I'm wondering if there have been any studies ...
5
votes
0answers
161 views

How hard is counting vertex covers / edge covers on the following graph class?

Let $G=(V,E)$ be a graph having all the following restrictions: Every vertex $v \in V$ has degree $4$. Every vertex $v \in V$ belongs to at least $2$ triangles. For every vertex $v \in V$, if $v$ ...