# Tag Info

### Status of PP-completeness of MAJ3SAT

Hopefully the following paper finally resolves this question: it says that MAJORITY 3SAT is in polynomial time. (And it proves a bunch of other unexpected results on related problems.) https://arxiv....
Accepted

### Counting the number of satisfying assignments in a POSITIVE CNF-SAT

This is still #P-complete . This problem is usually referred to as montone (#)SAT. Monotone #2-SAT is already #P-complete (this is equivalent to counting vertex covers of a graph).  Roth, Dan. "...

### Easy problems with hard counting versions

A very nice and simple example from Graph Theory is counting the number of Eularian circuits in an undirected graph. The decision version is easy (... and the Seven Bridges of Königsberg problem has ...

### Easy problems with hard counting versions

One interesting example from number theory is expressing a positive integer as a sum of four squares. This can be done relatively easily in random polynomial time (see my 1986 article with Rabin at ...

### Counting the number of satisfying assignments in a POSITIVE CNF-SAT

This problem is Monotone-SAT. It is #P-Complete under Cook Reductions. It is one of those problems that are "easy to decide but hard to count." I recommend the following paper. Self-Reducibility of ...
Accepted

### Why is the reduction from 3-SAT to 3-dimensional Matching Parsimonious?

You're right that the standard reduction from 3-SAT to 3D-matching (3DM) is not parsimonious. For the record, here's a sketch of a reduction that is parsimonious. It is obtained by composing ...
Accepted

### Complexity of counting matchings in a bipartite graph

The problem of counting such "imperfect" matchings in bipartite graphs is #P-complete. This has been proved by Les Valiant himself, on page 415 of the paper Leslie G. Valiant The Complexity of ...
Accepted

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

There exists a FPRAS (Fully Polynomial Randomized Approximation Scheme) for the problem of counting the words of length $n$ accepted by a NFA in the general case (without restricting to the acyclic ...

### More on PH in PP?

By the work of Klivans and van Melkebeek (which relativizes), if E = DTIME($2^{O(n)}$) does not have circuits with PP gates of size $2^{o(n)}$ then PH is in PP. The contrapositive says that if PH is ...
Accepted

### What's the complexity of counting odd nodes in graph?

Well, at least $\#\mathsf{P}$-hard. Given a SAT formula, construct a graph with two vertices, $v_x$ and $v_x'$, for every possible assignment of variables $\vec{x}$. If $x$ is a satisfying assignment ...
Accepted

### $⊕P$-completeness of $⊕2SAT$

It is shown to be $\oplus P$-complete by Faben: https://arxiv.org/abs/0809.1836 See Thm 3.5. Note that counting independent sets is same as counting solutions to monotone 2CNF.
Accepted

### Was counting complexity first introduced by Valiant in 1979?

Yes, the complexity class $\mathsf{\#P}$ is first introduced in Valiant's seminal paper "The complexity of computing the permanent." TCS, (1979). This is very clear. As for the terminology, strictly ...
Accepted

### Easy problems with hard counting versions

Here's a truly excellent example (I may be biased). Given a partially ordered set: a) does it have a linear extension (i.e., a total order compatible with the partial order)? Trivial: All posets ...
Accepted

### Concrete examples of $\sharp P_1$ complete problems? Self avoiding walks?

As someone with a long time interest in this complexity class, I don't believe there has been any significant work on $\#P_1$ since. (I was a Ph.D. student of Mitsu Ogihara, the second author of the ...
Accepted

### Is this a known problem, and is it #P-complete?

To Question 1: The graphs induced by such edge sets are known as pseudoforests. As your proof of the membership to #P implies, an edge set is the image of a selection function if and only if the graph ...

### Is #CYCLE #P-complete?

This is one of the problems (very briefly) discussed by Valiant, "The Complexity of Enumeration and Reliability Problems", SIAM J. Comput. 1979, doi:10.1137/0208032. See the mention of "elementary ...
Accepted

### Is counting simple cycles in $P$ for graphs of bounded tree width?

A simple cycle is a connected set where every vertex has degree 2. Then you have a formula SC(X) stating X (a set of edges) is a simple cycle. You can see many versions of Courcelle's theorem for ...
Accepted

### Counting avoiding improper 3-colorings

I don't know whether this problem has been studied but I think its #P-hardness should follow directly from the #CSP dichotomy established by Bulatov (Bulatov JACM'13), later simplified by Dyer and ...
### What is known about $\mathrm{NP}^{\mathrm{PP}}$?
Theorem 4.1 (ii) in J. Torán, Complexity Classes defined by Counting Quantifiers: $\exists \mathsf{PP} = \mathsf{NP}^{\mathsf{\# P}}$ (and thus $= \mathsf{NP}^{\mathsf{PP}}$). I also have a short, ...