The problem considered is to find a \(\lambda\)-optimal solution of a given integer linear programming problem, i.e., a solution which is optimal for some problem obtained from the original one via a perturbation of its right hand side (\(\lambda\) is the maximal coordinate difference). The authors give a quasi-polynomial algorithm for finding a \(\lambda\)-optimal solution of a linear 0-1 problem whose feasible set is \(\{\) x/ \(\sum_{j}\lambda_{ij}x_{ij}\leq b_ i\), \(\Sigma x_{ij}=1\}\).