  Nowadays the term 'sign problem' is used to identify two different problems.
The ideas to overcome the first type of the 'sign problem' of strongly
oscillating complex valued imtegrand in the Feynman path integrals comes from
Picard-Lefschetz theory and a complex version of Morse theory. The main idea is
to select Lefschetz thimbles as the cycle approaching the critical point at the
path-integration, where the imaginary part of the complex action stays
constant. Since the imaginary part of the action is constant on each thimble,
the sign problem disappears and the integral can be calculated much more
effectively. Here based on the Metropolis -- Hastings algorithm a new method of
calculations of the integral of the strongly oscillating integrands has been
prosed. Some simple test calculation and comparison with available analytical
results have been carried out. The second type of the 'sign problem' arises at
studies Fermi systems by path integral approach and is caused by the
requirement of antisymmetrization of the real valued matrix elements of the
density matrix. An explicit analytical expression for effective pair
pseudopotential in phase space has been discussed in Wigner formulation of
quantum mechanics. Obtained pseudopotential allow to account for Fermi
statistical effects as realizes the Pauli blocking of fermions due to the
repulsion between identical fermions, which prevents their occupation of same
phase space cell. To test this approach, calculations of the momentum
distribution function of the ideal system of Fermi particles have been
presented over a wide range of momentum and degeneracy parameter.
