9 research outputs found
The Fair Division of Hereditary Set Systems
We consider the fair division of indivisible items using the maximin shares
measure. Recent work on the topic has focused on extending results beyond the
class of additive valuation functions. In this spirit, we study the case where
the items form an hereditary set system. We present a simple algorithm that
allocates each agent a bundle of items whose value is at least times
the maximin share of the agent. This improves upon the current best known
guarantee of due to Ghodsi et al. The analysis of the algorithm is almost
tight; we present an instance where the algorithm provides a guarantee of at
most . We also show that the algorithm can be implemented in polynomial
time given a valuation oracle for each agent.Comment: 22 pages, 1 figure, full version of WINE 2018 submissio
Simplification and Improvement of MMS Approximation
We consider the problem of fairly allocating a set of indivisible goods among
agents with additive valuations, using the popular fairness notion of
maximin share (MMS). Since MMS allocations do not always exist, a series of
works provided existence and algorithms for approximate MMS allocations. The
current best approximation factor, for which the existence is known, is
[Garg and Taki, 2021]. Most of these results
are based on complicated analyses, especially those providing better than
factor. Moreover, since no tight example is known of the Garg-Taki algorithm,
it is unclear if this is the best factor of this approach. In this paper, we
significantly simplify the analysis of this algorithm and also improve the
existence guarantee to a factor of . For small , this provides a noticeable improvement.
Furthermore, we present a tight example of this algorithm, showing that this
may be the best factor one can hope for with the current techniques
Randomized and Deterministic Maximin-share Approximations for Fractionally Subadditive Valuations
We consider the problem of guaranteeing maximin-share (MMS) when allocating a
set of indivisible items to a set of agents with fractionally subadditive (XOS)
valuations. For XOS valuations, it has been previously shown that for some
instances no allocation can guarantee a fraction better than of
maximin-share to all the agents. Also, a deterministic allocation exists that
guarantees of the maximin-share of each agent. Our results involve
both deterministic and randomized allocations. On the deterministic side, we
improve the best approximation guarantee for fractionally subadditive
valuations to . We develop new ideas on allocating large items
in our allocation algorithm which might be of independent interest.
Furthermore, we investigate randomized algorithms and the Best-of-both-worlds
fairness guarantees. We propose a randomized allocation that is -MMS
ex-ante and -MMS ex-post for XOS valuations. Moreover, we prove an upper
bound of on the ex-ante guarantee for this class of valuations
Improving Approximation Guarantees for Maximin Share
We consider fair division of a set of indivisible goods among agents with
additive valuations using the desirable fairness notion of maximin share (MMS).
MMS is the most popular share-based notion, in which an agent finds an
allocation fair to her if she receives goods worth at least her MMS value. An
allocation is called MMS if all agents receive their MMS values. However, since
MMS allocations do not always exist, the focus shifted to investigating its
ordinal and multiplicative approximations. In the ordinal approximation, the
goal is to show the existence of -out-of- MMS allocations (for the
smallest possible ). A series of works led to the state-of-the-art factor
of [HSSH21]. We show that -out-of- MMS allocations always exist. In the multiplicative approximation,
the goal is to show the existence of -MMS allocations (for the largest
possible ) which guarantees each agent at least times her
MMS value. A series of works in the last decade led to the state-of-the-art
factor of [AG23]. We introduce a
general framework of -MMS that guarantees
fraction of agents times their MMS values and the remaining
fraction of agents times their MMS values. The -MMS captures both ordinal and multiplicative approximations as
its special cases. We show that -MMS
allocations always exist. Furthermore, since we can choose the
fraction of agents arbitrarily in our algorithm, this
implies (using ) the existence of a randomized allocation
that gives each agent at least 3/4 times her MMS value (ex-post) and at least
times her MMS value in expectation
(ex-ante)
Breaking the Barrier for Approximate Maximin Share
We study the fundamental problem of fairly allocating a set of indivisible
goods among agents with additive valuations using the desirable fairness
notion of maximin share (MMS). MMS is the most popular share-based notion, in
which an agent finds an allocation fair to her if she receives goods worth at
least her MMS value. An allocation is called MMS if all agents receive at least
their MMS value. However, since MMS allocations need not exist when , a
series of works showed the existence of approximate MMS allocations with the
current best factor of . The recent work by
Akrami et al. showed the limitations of existing approaches and proved that
they cannot improve this factor to . In this paper, we bypass
these barriers to show the existence of -MMS
allocations by developing new reduction rules and analysis techniques
Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results
Fair resource allocation is an important problem in many real-world
scenarios, where resources such as goods and chores must be allocated among
agents. In this survey, we delve into the intricacies of fair allocation,
focusing specifically on the challenges associated with indivisible resources.
We define fairness and efficiency within this context and thoroughly survey
existential results, algorithms, and approximations that satisfy various
fairness criteria, including envyfreeness, proportionality, MMS, and their
relaxations. Additionally, we discuss algorithms that achieve fairness and
efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study
the computational complexity of these algorithms, the likelihood of finding
fair allocations, and the price of fairness for each fairness notion. We also
cover mixed instances of indivisible and divisible items and investigate
different valuation and allocation settings. By summarizing the
state-of-the-art research, this survey provides valuable insights into fair
resource allocation of indivisible goods and chores, highlighting computational
complexities, fairness guarantees, and trade-offs between fairness and
efficiency. It serves as a foundation for future advancements in this vital
field