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Is deciding whether changing one entry decreases the permanent of a matrix in the polynomial...

Your problem is equivalent to testing, given $M$, whether $PER(M) > 0$. Proof: Assume you are given $M$ and you want to decide whether $PER(M) > 0$. We construct $M'$ as follows: \begin{bmatrix} …
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9 votes
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Complexity of computing generalised determinants. (P - #P transition)

Nicolas de Rugy-Altherre studied the complexity of one of them, the Fermionant, and proves other $VNP$-completeness results for such functions in "Determinant versus Permanent: salvation via generalization …
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6 votes

Easy problems with hard counting versions

Concerning your second question, problems such as Monotone-2-SAT (deciding of the satisfiability of a CNF-formula having at most 2 positive literals by clause) is completely trivial (you just have to …
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