5 research outputs found
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
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)
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
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
Fair and Efficient Allocation of Indivisible Chores with Surplus
We study fair division of indivisible chores among agents with additive
disutility functions. Two well-studied fairness notions for indivisible items
are envy-freeness up to one/any item (EF1/EFX) and the standard notion of
economic efficiency is Pareto optimality (PO). There is a noticeable gap
between the results known for both EF1 and EFX in the goods and chores
settings. The case of chores turns out to be much more challenging. We reduce
this gap by providing slightly relaxed versions of the known results on goods
for the chores setting. Interestingly, our algorithms run in polynomial time,
unlike their analogous versions in the goods setting.
We introduce the concept of surplus which means that up to more
chores are allocated to the agents and each of them is a copy of an original
chore. We present a polynomial-time algorithm which gives EF1 and PO
allocations with surplus.
We relax the notion of EFX slightly and define tEFX which requires that the
envy from agent to agent is removed upon the transfer of any chore from
the 's bundle to 's bundle. We give a polynomial-time algorithm that in
the chores case for agents returns an allocation which is either
proportional or tEFX. Note that proportionality is a very strong criterion in
the case of indivisible items, and hence both notions we guarantee are
desirable