Computing the permanent of (some) complex matrices

where Sn is the symmetric group of permutations of the set {1, . . . , n}. The problem of efficient computation of the permanent has attracted a lot of attention. It is #P -hard already for 0-1 matrices [Va79], but a fully polynomial randomized approximation scheme, based on the Markov Chain Monte Carlo approach, is constructed for all non-negative matrices [J+04]. A deterministic polynomial time algorithm based on matrix scaling for computing the permanent of non-negative matrices within a factor of e is constructed in [L+00] and the bound was recently improved to 2 in [GS13]. An approach based on the idea of “correlation decay” from statistical physics results in a deterministic polynomial time algorithm approximating perA within a factor of (1 + ǫ) for any ǫ > 0, fixed in advance, if A is the adjacency matrix of a constant degree expander [GK10]. There is also interest in computing permanents of complex matrices [AA13]. The well-known Ryser’s algorithm (see, for example, Chapter 7 of [Mi78]) computes the