The Bethe permanent of a nonnegative square matrix 
is an approximation to its
permanent defined by maximizing a product over doubly
stochastic matrices 
supported on the nonzero entries of 
:
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Factors at the endpoints are interpreted by continuity. If the support has no perfect matching, the Bethe permanent is defined
to be zero. The Bethe permanent is a lower bound for the permanent,
and the ratio of the permanent to the Bethe permanent
is at most 
for an 
matrix with positive permanent (Dong and Jain 2026).
The bipartite support graph has one vertex for each row and column and an edge for each positive entry. Dong and Jain (2026) proved the sharp improvement
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when the support graph has girth at least an even integer 
. Disjoint cycles of length 
attain equality when 
divides 
. Their proof was co-developed with
GPT-5.6 Sol and checked by the authors. Independent external review had not been
reported as of Sep. 7, 2026.
