Bounding the Bethe and the Degree-M Bethe Permanents

In [1], it was conjectured that the permanent of a P-lifting θ of a matrix θ of degree M is less than or equal to the M th power of the permanent perm(θ), i.e., perm(θ) 6 perm(θ) and, consequently, that the degree-M Bethe permanent permM,B(θ) of a matrix θ is less than or equal to the permanent perm(θ) of θ, i.e., permM,B(θ) 6 perm(θ). In this paper, we prove these related conjectures and show in addition a few 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) 6 perm(θ) on the Bethe permanent of the base matrix θ that uses only the combinatorial definition of the Bethe permanent (the first proof was given by Gurvits in [2]).