A generalized permanent estimator and its application in computing multi-homogeneous Bézout Number

Abstract

The permanent of a matrix has many applications in many fields. Its computation is #P-complete. The computation of exact permanent large-scale matrices is very costly in terms of memory and time. There is a real need for an efficient method to deal well with such situations. This study designs a general algorithm for estimating the permanents of the complex square or non-square matrices. We prove that the Multi-Homogeneous Bézout Number (MHBN) can be estimated efficiently using the new algorithm. Further, a proposition that provides some analytic results is presented and proved. The analytic results show the effectiveness and the efficiency of our algorithm over some recent methods. Furthermore, with the new algorithm we can control the accuracy as we need. A large amount of numerical results are presented in this study. By applying the algorithm that estimates MHBN we extend the applicability of the algorithm.