A set of objects, some goods and some bads , 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  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  division? We focus on Pareto-optimal, envy-free and/or proportional allocations. 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. The problem becomes computationally hard for degenerate instances, where the agents' valuations are aligned for many objects.
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? We focus on Pareto-optimal, envy-free and/or proportional allocations. 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. The problem becomes computationally hard for degenerate instances, where agents' valuations are aligned for many objects.
R
set
collection
R
goods and some bads
of which are good and some are bad
A
a
A
and efficient
R
---
--
R
--- a fair and
-- a fair
D
the