Unleashing the power of Schrijver's permanental inequality with the help of the Bethe Approximation

Let A ∈ Ωn be doubly-stochastic n × n matrix. Alexander Schrijver proved in 1998 the following remarkable inequality per(Ã) ≥ ∏ 1≤i,j≤n (1−A(i, j)); Ã(i, j) =: A(i, j)(1−A(i, j)), 1 ≤ i, j ≤ n (1) We prove in this paper the following generalization (or just clever reformulation) of (1): For all pairs of n × n matrices (P,Q), where P is nonnegative and Q is doublystochastic log(per(P )) ≥ ∑ 1≤i,j≤n log(1−Q(i, j))(1−Q(i, j))− ∑ 1≤i,j≤n Q(i, j) log ( Q(i, j) P (i, j) ) (2) The main co rollary of (2) is the following inequality for doubly-stochastic matrices: per(A) F (A) ≥ 1;F (A) =: ∏ 1≤i,j≤n (1−A(i, j))1−A(i,j) . We use this inequality to prove Friedland’s conjecture on monomerdimer entropy, so called Asymptotic Lower Matching Conjecture We present explicit doubly-stochastic n×n matrices A with the ratio per(A) F (A) = √ 2 n and conjecture that max A∈Ωn per(A) F (A) ≈ (√ 2 )n . If true, it would imply a deterministic poly-time algorithm to approximate the permanent of n× n nonnegative matrices within the relative factor (√ 2 )n . ∗[email protected]. Los Alamos National Laboratory, Los Alamos, NM.