  We discuss schemes for exact and approximate computations of permanents, and
compare them with each other. Specifically, we analyze the Belief Propagation
(BP) approach and its Fractional Belief Propagation (FBP) generalization for
computing the permanent of a non-negative matrix. Known bounds and conjectures
are verified in experiments, and some new theoretical relations, bounds and
conjectures are proposed. The Fractional Free Energy (FFE) functional is
parameterized by a scalar parameter $\gamma\in[-1;1]$, where $\gamma=-1$
corresponds to the BP limit and $\gamma=1$ corresponds to the exclusion
principle (but ignoring perfect matching constraints) Mean-Field (MF) limit.
FFE shows monotonicity and continuity with respect to $\gamma$. For every
non-negative matrix, we define its special value $\gamma_*\in[-1;0]$ to be the
$\gamma$ for which the minimum of the $\gamma$-parameterized FFE functional is
equal to the permanent of the matrix, where the lower and upper bounds of the
$\gamma$-interval corresponds to respective bounds for the permanent. Our
experimental analysis suggests that the distribution of $\gamma_*$ varies for
different ensembles but $\gamma_*$ always lies within the $[-1;-1/2]$ interval.
Moreover, for all ensembles considered the behavior of $\gamma_*$ is highly
distinctive, offering an emprirical practical guidance for estimating
permanents of non-negative matrices via the FFE approach.
