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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?

Are there problems defined on graphs, such as counting 2-factors, Hamiltonian cycles, connected spanning subgraphs etc., that are in $\#P$ and remain hard for grid graphs? Since there seem to be multi …

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 instanc …

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 questi …

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 …

Corollary 3.6 in the UniqueSAT paper by Valiant and Vazirani [1] states: For any $\varepsilon > 0$ there is a randomized polynomial-time TM with a SAT oracle, which given a SAT formula $f$ outputs a …

This paper shows an exact mapping from reliability to exact model counting. From that point on, exact counters (like miniC2D) can be used to compute reliability. Not sure if useful runtime bounds exis …

Commonly occurring #P problems (e.g., #SAT) are often studied in the regime where instances with $n$ Boolean variables have $2^{\Theta(n)}$ solutions, in part because instances with $O(1)$ solutions c …