Modified Bethe Permanent of a Nonnegative Matrix

Currently the best deterministic polynomial-time algorithm for approximating the permanent of a non-negative matrix is based on minimizing the Bethe free energy function of a certain normal factor graph (NFG). In order to improve the approximation guarantee, we propose a modified NFG with fewer cycles, but still manageable function-node complexity; we call the approximation obtained by minimizing the function of the modified normal factor graph the modified Bethe permanent. For nonnegative matrices of size $3\times 3$, we give a tight characterization of the modified Bethe permanent. For non-negative matrices of size $n\times n$ with $n\geq 3$, we present a partial characterization, along with promising numerical results. The analysis of the modified NFG is also interesting because of its tight connection to an NFG that is used for approximating a permanent-like quantity in quantum information processing.