We consider fair division of a set of indivisible goods among n agents with additive valuations using the 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 (1-out-of-n) MMS value. An allocation is called MMS if all agents receive their MMS values. However, since MMS allocations do not always exist [Kurokawa et al., JACM'18], the focus shifted to investigating its ordinal and multiplicative approximations.

We consider the problem of fairly allocating a set of indivisible goods among n 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 Garg-Taki algorithm gives the current best approximation factor of (3/4 + 1/12n). Most of these results are based on complicated analyses, especially those providing better than 2/3 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 (3/4 + min(1/36, 3/(16n-4))). For small n, 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.

We study fair allocation of indivisible items, both goods and chores, under the popular fairness notion of maximin share (MMS). The problem is well-studied when there are only goods (or chores), where a PTAS to compute the MMS values of agents is well-known. In contrast, for the mixed manna, a recent result showed that finding even an approximate MMS value of an agent up to any approximation factor in (0,1] is NP-hard for general instances. In this paper, we complement the hardness result by obtaining a PTAS to compute the MMS value when its absolute value is at least 1/p times either the total value of all the goods or total cost of all the chores, for some constant p valued at least 1.

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We study the problem of fair allocation of m indivisible items among n agents with additive valuations using the popular notion of maximin share (MMS) as our measure of fairness. An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist [5, 7], a series of remarkable work [1-3, 6, 7] provided 2/3 approximation algorithms in which each agent receives a bundle worth at least 2/3 times her maximin share. More recently, [4] showed the existence of 3/4 MMS allocations and a PTAS to find a 3/4 - ε MMS allocation. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain.