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As an effective model corresponding to $Z_3$-symmetric QCD ($Z_3$-QCD), weconstruct a $Z_3$-symmetric effective Polyakov-line model ($Z_3$-EPLM) by usingthe logarithmic fermion effective action. Since $Z_3$-QCD tends to QCD in thezero temperature limit, $Z_3$-EPLM also agrees with the ordinary effectivePolyakov-line model (EPLM) there; note that ordinary EPLM does not possess$Z_3$ symmetry. Our main purpose is to discuss a sign problem appearing in$Z_3$-EPLM. The action of $Z_3$-EPLM is real, when the Polyakov line is notonly real but also its $Z_3$ images. This suggests that the sign problembecomes milder in $Z_3$-EPLM than in EPLM. In order to confirm this suggestion,we do lattice simulations for both EPLM and $Z_3$-EPLM by using the reweightingmethod with the phase quenched approximation. In the low-temperature region,the sign problem is milder in $Z_3$-EPLM than in EPLM. We also propose a newreweighting method. This makes the sign problem very weak in $Z_3$-EPLM.

Sign problem in $Z_3$-symmetric effective Polyakov-line model