201 research outputs found
Fair Division of a Graph
We consider fair allocation of indivisible items under an additional
constraint: there is an undirected graph describing the relationship between
the items, and each agent's share must form a connected subgraph of this graph.
This framework captures, e.g., fair allocation of land plots, where the graph
describes the accessibility relation among the plots. We focus on agents that
have additive utilities for the items, and consider several common fair
division solution concepts, such as proportionality, envy-freeness and maximin
share guarantee. While finding good allocations according to these solution
concepts is computationally hard in general, we design efficient algorithms for
special cases where the underlying graph has simple structure, and/or the
number of agents -or, less restrictively, the number of agent types- is small.
In particular, despite non-existence results in the general case, we prove that
for acyclic graphs a maximin share allocation always exists and can be found
efficiently.Comment: 9 pages, long version of accepted IJCAI-17 pape
Chore division on a graph
The paper considers fair allocation of indivisible nondisposable items that
generate disutility (chores). We assume that these items are placed in the
vertices of a graph and each agent's share has to form a connected subgraph of
this graph. Although a similar model has been investigated before for goods, we
show that the goods and chores settings are inherently different. In
particular, it is impossible to derive the solution of the chores instance from
the solution of its naturally associated fair division instance. We consider
three common fair division solution concepts, namely proportionality,
envy-freeness and equitability, and two individual disutility aggregation
functions: additive and maximum based. We show that deciding the existence of a
fair allocation is hard even if the underlying graph is a path or a star. We
also present some efficiently solvable special cases for these graph
topologies
Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints
We study the problem of allocating indivisible items to agents with additive
valuations, under the additional constraint that bundles must be connected in
an underlying item graph. Previous work has considered the existence and
complexity of fair allocations. We study the problem of finding an allocation
that is Pareto-optimal. While it is easy to find an efficient allocation when
the underlying graph is a path or a star, the problem is NP-hard for many other
graph topologies, even for trees of bounded pathwidth or of maximum degree 3.
We show that on a path, there are instances where no Pareto-optimal allocation
satisfies envy-freeness up to one good, and that it is NP-hard to decide
whether such an allocation exists, even for binary valuations. We also show
that, for a path, it is NP-hard to find a Pareto-optimal allocation that
satisfies maximin share, but show that a moving-knife algorithm can find such
an allocation when agents have binary valuations that have a non-nested
interval structure.Comment: 21 pages, full version of paper at AAAI-201
The Price of Connectivity in Fair Division
We study the allocation of indivisible goods that form an undirected graph
and quantify the loss of fairness when we impose a constraint that each agent
must receive a connected subgraph. Our focus is on well-studied fairness
notions including envy-freeness and maximin share fairness. We introduce the
price of connectivity to capture the largest gap between the graph-specific and
the unconstrained maximin share, and derive bounds on this quantity which are
tight for large classes of graphs in the case of two agents and for paths and
stars in the general case. For instance, with two agents we show that for
biconnected graphs it is possible to obtain at least of the maximin share
with connected allocations, while for the remaining graphs the guarantee is at
most . In addition, we determine the optimal relaxation of envy-freeness
that can be obtained with each graph for two agents, and characterize the set
of trees and complete bipartite graphs that always admit an allocation
satisfying envy-freeness up to one good (EF1) for three agents. Our work
demonstrates several applications of graph-theoretic tools and concepts to fair
division problems
Chore division on a graph
Le PDF est une version non publiée datant de 2018.International audienceThe paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent’s share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies
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? 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.Comment: Add experiments with Spliddit.org dat
- …