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We do not have a deterministic constant factor approximation scheme for general $n\times n$ $0/1$ permanent.

What is the best factor in deterministic approximation schemes if we only care counting bipartite perfect matching with average degree in $[2,3]$ and max degree $3$?

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Dagum and Luby show (using a construction credited to Dahlhaus and Karpinski) how to construct, given a bipartite graph $G$, a bipartite graph $G'$ of maximum degree $3$ such that $G'$ has exactly as many perfect matchings as $G$ (see Theorem 6.2.). Then from $G'$ you can construct a graph $G''$ with average degree arbitrarily close to $2$ and twice as many perfect matchings as $G'$, as I explained here. Both constructions are in polynomial time. Therefore, the best polytime deterministic approximation to the number of perfect matchings of a graph of maximum degree 3, and average degree in $[2,3]$ is equal to the best polytime deterministic approximation to the permanent of a 0-1 matrix. As far as I know, this is the factor $2^n$ approximation achieved by Gurvits and Samorodnitsky.

(It may be helpful to note that a $\exp(n^\varepsilon)$ approximation for $\varepsilon < 1$ would imply an FPTAS.)

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  • $\begingroup$ do you know which page they talk about this construction in Dagum and Luby and also how many vertices are used in $G'$? $\endgroup$ – Turbo Jul 24 '17 at 5:53
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    $\begingroup$ As I said in my answer, it's Theorem 6.2. The proof is on page 299. $G'$ has $2m - n/2$ vertices, where $m$ is the number of edges of $G$ and $n$ is the number of vertices of $G$. $\endgroup$ – Sasho Nikolov Jul 24 '17 at 6:13

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