# Questions tagged [counting-complexity]

How hard is counting the number of solutions?

51 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
622 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". ...
707 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 ...
977 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 ...
125 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, ...
293 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 ...
462 views

### What is the complexity of counting the number of solutions of a P-Space Complete problem? How about higher complexity classes?

I guess it would be called #P-Space but I have found only one article vaguely mentioning it. How about the counting version of EXP-TIME-Complete, NEXP-Complete as well as EXP-SPACE-Complete problems? ...
134 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 ...
139 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 ...
607 views

### Complexity of Exactly $A$-SAT

Let define the "Exactly $A$-SAT" problem : Given a CNF formula $F$ and a set $A$ of positive integers, is it true that the number of models of $F$ is in $A$? What is the complexity of Exactly $A$-...
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 ...
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 ...
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 ...
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 ...
143 views

107 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 6....
72 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 $\#P-Complete$, ...
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....
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 ...
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. ...
461 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 ...
### 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 $\Bbb F_{3^t}$ ...