Anari's Bethe permanent conjecture

For every nonnegative $n\times n$ matrix $A$ whose bipartite support graph has girth at least an even integer $g\ge4$, Dong and Jain prove the sharp inequality $$\operatorname{Bethe}(A)\le\operatorname{per}(A)\le2^{2n/g}\operatorname{Bethe}(A).$$ The factor $2^{2n/g}$ is optimal whenever $g\mid2n$, attained by matrices whose support graphs are disjoint unions of $g$-cycles. Thus the result confirms Anari's conjecture, recovers the sharp $2^{n/2}$ universal Anari--Rezaei bound when $g=4$, and approaches exactness as the support-graph girth tends to infinity.

For every nonnegative $n\times n$ matrix $A$ whose bipartite support graph has girth at least an even integer $g\ge4$, Dong and Jain prove the sharp inequality $$\operatorname{Bethe}(A)\le\operatorname{per}(A)\le2^{2n/g}\operatorname{Bethe}(A).$$ The factor $2^{2n/g}$ is optimal whenever $g\mid2n$, attained by matrices whose support graphs are disjoint unions of $g$-cycles. Thus the result confirms Anari's conjecture, recovers the sharp $2^{n/2}$ universal Anari--Rezaei bound when $g=4$, and approaches exactness as the support-graph girth tends to infinity.