Bethe and M-Bethe Permanent Inequalities

In [1], it was conjectured that the permanent of a P-lifting θ^↑P of a matrix θ of degree M is less than or equal to the Mth power of the permanent perm(θ), i.e., perm(θ^↑P) ≤ perm(θ)^M and, consequently, that the degree-M Bethe permanent perm_M,B(θ) of a matrix θ is less than or equal to the permanent perm(θ) of θ, i.e., perm_M,B(θ) ≤ perm(θ). In this paper, we prove these related conjectures and show some properties of the permanent of block matrices that are lifts of a matrix. As a corollary, we obtain an alternative proof of the inequality perm_B(θ) ≤ perm(θ) on the Bethe permanent of the base matrix θ, which, in contrast to the one given in [2], uses only the combinatorial definition of the Bethe-permanent. The results have implications in coding theory. Since a P-lifting corresponds to an M-graph cover and thus to a protograph-based LDPC code, the results may help explain the performance of these codes.