3,116 research outputs found
On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources
We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study envy-free allocation of chores and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting a claim in the existing literature. A modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (where each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. [Bei et al., 2021] for indivisible goods and divisible cake
Fair Division of Mixed Divisible and Indivisible Goods
We study the problem of fair division when the resources contain both
divisible and indivisible goods. Classic fairness notions such as envy-freeness
(EF) and envy-freeness up to one good (EF1) cannot be directly applied to the
mixed goods setting. In this work, we propose a new fairness notion
envy-freeness for mixed goods (EFM), which is a direct generalization of both
EF and EF1 to the mixed goods setting. We prove that an EFM allocation always
exists for any number of agents. We also propose efficient algorithms to
compute an EFM allocation for two agents and for agents with piecewise
linear valuations over the divisible goods. Finally, we relax the envy-free
requirement, instead asking for -envy-freeness for mixed goods
(-EFM), and present an algorithm that finds an -EFM
allocation in time polynomial in the number of agents, the number of
indivisible goods, and .Comment: Appears in the 34th AAAI Conference on Artificial Intelligence
(AAAI), 202
Guaranteeing Envy-Freeness under Generalized Assignment Constraints
We study fair division of goods under the broad class of generalized
assignment constraints. In this constraint framework, the sizes and values of
the goods are agent-specific, and one needs to allocate the goods among the
agents fairly while further ensuring that each agent receives a bundle of total
size at most the corresponding budget of the agent. Since, in such a constraint
setting, it may not always be feasible to partition all the goods among the
agents, we conform -- as in recent works -- to the construct of charity to
designate the set of unassigned goods. For this allocation framework, we obtain
existential and computational guarantees for envy-free (appropriately defined)
allocation of divisible and indivisible goods, respectively, among agents with
individual, additive valuations for the goods.
We deem allocations to be fair by evaluating envy only with respect to
feasible subsets. In particular, an allocation is said to be feasibly envy-free
(FEF) iff each agent prefers its bundle over every (budget) feasible subset
within any other agent's bundle (and within the charity). The current work
establishes that, for divisible goods, FEF allocations are guaranteed to exist
and can be computed efficiently under generalized assignment constraints.
In the context of indivisible goods, FEF allocations do not necessarily
exist, and hence, we consider the fairness notion of feasible envy-freeness up
to any good (FEFx). We show that, under generalized assignment constraints, an
FEFx allocation of indivisible goods always exists. In fact, our FEFx result
resolves open problems posed in prior works. Further, for indivisible goods and
under generalized assignment constraints, we provide a pseudo-polynomial time
algorithm for computing FEFx allocations, and a fully polynomial-time
approximation scheme (FPTAS) for computing approximate FEFx allocations.Comment: 29 page
An EF2X Allocation Protocol for Restricted Additive Valuations
We study the problem of fairly allocating a set of indivisible goods to aset of agents. Envy-freeness up to any good (EFX) criteria -- whichrequires that no agent prefers the bundle of another agent after removal of anysingle good -- is known to be a remarkable analogous of envy-freeness when theresource is a set of indivisible goods. In this paper, we investigate EFXnotion for the restricted additive valuations, that is, every good has somenon-negative value, and every agent is interested in only some of the goods. We introduce a natural relaxation of EFX called EFkX which requires that noagent envies another agent after removal of any goods. Our maincontribution is an algorithm that finds a complete (i.e., no good is discarded)EF2X allocation for the restricted additive valuations. In our algorithm wedevise new concepts, namely "configuration" and "envy-elimination" that mightbe of independent interest. We also use our new tools to find an EFX allocation for restricted additivevaluations that discards at most goods. This improvesthe state of the art for the restricted additive valuations by a factor of .<br
Finding Fair and Efficient Allocations
We study the problem of allocating a set of indivisible goods among a set of
agents in a fair and efficient manner. An allocation is said to be fair if it
is envy-free up to one good (EF1), which means that each agent prefers its own
bundle over the bundle of any other agent up to the removal of one good. In
addition, an allocation is deemed efficient if it satisfies Pareto optimality
(PO). While each of these well-studied properties is easy to achieve
separately, achieving them together is far from obvious. Recently, Caragiannis
et al. (2016) established the surprising result that when agents have additive
valuations for the goods, there always exists an allocation that simultaneously
satisfies these two seemingly incompatible properties. Specifically, they
showed that an allocation that maximizes the Nash social welfare (NSW)
objective is both EF1 and PO. However, the problem of maximizing NSW is
NP-hard. As a result, this approach does not provide an efficient algorithm for
finding a fair and efficient allocation.
In this paper, we bypass this barrier, and develop a pseudopolynomial time
algorithm for finding allocations that are EF1 and PO; in particular, when the
valuations are bounded, our algorithm finds such an allocation in polynomial
time. Furthermore, we establish a stronger existence result compared to
Caragiannis et al. (2016): For additive valuations, there always exists an
allocation that is EF1 and fractionally PO.
Another contribution of our work is to show that our algorithm provides a
polynomial-time 1.45-approximation to the NSW objective. This improves upon the
best known approximation ratio for this problem (namely, the 2-approximation
algorithm of Cole et al. (2017)). Unlike many of the existing approaches, our
algorithm is completely combinatorial.Comment: 40 pages. Updated versio
Fairly Allocating Many Goods with Few Queries
We investigate the query complexity of the fair allocation of indivisible
goods. For two agents with arbitrary monotonic valuations, we design an
algorithm that computes an allocation satisfying envy-freeness up to one good
(EF1), a relaxation of envy-freeness, using a logarithmic number of queries. We
show that the logarithmic query complexity bound also holds for three agents
with additive valuations. These results suggest that it is possible to fairly
allocate goods in practice even when the number of goods is extremely large. By
contrast, we prove that computing an allocation satisfying envy-freeness and
another of its relaxations, envy-freeness up to any good (EFX), requires a
linear number of queries even when there are only two agents with identical
additive valuations
Fair Division with a Secretive Agent
We study classic fair-division problems in a partial information setting.
This paper respectively addresses fair division of rent, cake, and indivisible
goods among agents with cardinal preferences. We will show that, for all of
these settings and under appropriate valuations, a fair (or an approximately
fair) division among n agents can be efficiently computed using only the
valuations of n-1 agents. The nth (secretive) agent can make an arbitrary
selection after the division has been proposed and, irrespective of her choice,
the computed division will admit an overall fair allocation.
For the rent-division setting we prove that the (well-behaved) utilities of
n-1 agents suffice to find a rent division among n rooms such that, for every
possible room selection of the secretive agent, there exists an allocation (of
the remaining n-1 rooms among the n-1 agents) which ensures overall envy
freeness (fairness). We complement this existential result by developing a
polynomial-time algorithm that finds such a fair rent division under
quasilinear utilities.
In this partial information setting, we also develop efficient algorithms to
compute allocations that are envy-free up to one good (EF1) and
epsilon-approximate envy free. These two notions of fairness are applicable in
the context of indivisible goods and divisible goods (cake cutting),
respectively. This work also addresses fairness in terms of proportionality and
maximin shares. Our key result here is an efficient algorithm that, even with a
secretive agent, finds a 1/19-approximate maximin fair allocation (of
indivisible goods) under submodular valuations of the non-secretive agents.
One of the main technical contributions of this paper is the development of
novel connections between different fair-division paradigms, e.g., we use our
existential results for envy-free rent-division to develop an efficient EF1
algorithm.Comment: 27 page
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