Before revision:  We show that, for a generic instance of the problem --- all instances except of a zero-measure set of degenerate problems --- a fair and Pareto-optimal division with the smallest possible number of shared objects can be found in polynomial time, assuming that the number of agents is fixed.
After revision:  We show that, for a generic instance of the problem --- all instances except of a zero-measure set of degenerate problems -- a fair Pareto-optimal division with the smallest possible number of shared objects can be found in polynomial time, assuming that the number of agents is fixed.