Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Efficient Fair Division with Minimal Sharing   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.

23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th