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We investigate the QCD phase diagram in the strong-coupling lattice QCD withfluctuation and $1/g^2$ effects by using the auxiliary field Monte-Carlosimulations. The complex phase of the Fermion determinant at finite chemicalpotential is found to be suppressed by introducing a complex shift of integralpath for one of the auxiliary fields, which corresponds to introducing arepulsive vector mean field for quarks. The obtained phase diagram in thechiral limit shows suppressed $T_c$ in the second order phase transition regioncompared with the strong-coupling limit results. We also argue that we canapproximately guess the statistical weight cancellation from the complex phasein advance in the case where the complex phase distribution is Gaussian. Wedemonstrate that correct expectation values are obtained by using this guess inthe importance sampling (preweighting).

Preweighting method in Monte-Carlo sampling with complex action --- Strong-Coupling Lattice QCD with $1/g^2$ corrections, as an example ---