  A collection of objects, some of which are good and some are bad, is to be
divided fairly among agents with different tastes, modeled by additive utility
functions. If the objects cannot be shared, so that each of them must be
entirely allocated to a single agent, then a fair division may not exist. What
is the smallest number of objects that must be shared between two or more
agents in order to attain a fair and efficient division?
  In this paper, fairness is understood as proportionality or envy-freeness,
and efficiency, as fractional Pareto-optimality. We show that, for a generic
instance of the problem (all instances except a zero-measure set of degenerate
problems), a fair fractionally 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. The problem becomes computationally hard
for degenerate instances, where agents' valuations are aligned for many
objects.
