arXiv subjects

Setareh Taki

Publications and source records attributed to Setareh Taki.

4 recordsLinked to original sources

arXiv2023

Improving Approximation Guarantees for Maximin Share

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, the focus shifted to investigating its ordinal and multiplicative approximations. In the ordinal approximation, the goal is to show the existence of $1$-out-of-$d$ MMS allocations (for the smallest possible $d>n$). A series of works led to the state-of-the-art factor of $d=\lfloor3n/2\rfloor$ [Hosseini et al.'21]. We show that $1$-out-of-$4\lceil n/3\rceil$ MMS allocations always exist, thereby improving the state-of-the-art of ordinal approximation. In the multiplicative approximation, the goal is to show the existence of $\alpha$-MMS allocations (for the largest possible $\alpha < 1$), which guarantees each agent at least $\alpha$ times her MMS value. We introduce a general framework of "approximate MMS with agent priority ranking". An allocation is said to be $T$-MMS, for a non-increasing sequence $T = (\tau_1, \ldots, \tau_n)$ of numbers, if the agent at rank $i$ in the order gets a bundle of value at least $\tau_i$ times her MMS value. This framework captures both ordinal approximation and multiplicative approximation as special cases. We show the existence of $T$-MMS allocations where $\tau_i \ge \max(\frac{3}{4} + \frac{1}{12n}, \frac{2n}{2n+i-1})$ for all $i$. Furthermore, we can get allocations that are $(\frac{3}{4} + \frac{1}{12n})$-MMS ex-post and $(0.8253 + \frac{1}{36n})$-MMS ex-ante. We also prove that our algorithm does not give better than $(0.8631 + \frac{1}{2n})$-MMS ex-ante.

cs.GT↗

arXiv2023

Simplification and Improvement of MMS Approximation

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 $(\frac{3}{4} + \frac{1}{12n})$. 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 $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. 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.

cs.GT↗

arXiv2021

An Improved Approximation Algorithm for Maximin Shares

Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case where m indivisible items need to be divided among n agents with additive valuations using the popular fairness notion of maximin share (MMS). 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, a series of work provided approximation algorithms for a 2/3-MMS allocation in which each agent receives a bundle worth at least 2/3 times her maximin share. More recently, Ghodsi et al. [EC'2018] showed the existence of a 3/4-MMS allocation and a PTAS to find a (3/4-ε)-MMS allocation for an ε> 0. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain. In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a 3/4-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial-time algorithm to find a 3/4-MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a (3/4 + 1/(12n))-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that 3/4 was the best factor known for n> 4.

cs.GT↗

arXiv2021

Indivisible Mixed Manna: On the Computability of MMS + PO Allocations

In this paper we initiate the study of finding fair and efficient allocations of an indivisible mixed manna: Divide m indivisible items among n agents under the fairness notion of maximin share (MMS) and the efficiency notion of Pareto optimality (PO). A mixed manna allows an item to be a good for some agents and a chore for others. The problem of finding $α$-MMS allocation for the (near) best $α\in(0,1]$ for which it exists, remains unresolved even for a goods manna with constantly many agents, while the problem of finding $α$-MMS+PO allocation is unexplored for any $α\in(0,1]$. We make significant progress on the above questions for a mixed manna. First, we show that for any $α>0$, an $α$-MMS allocation may not always exist, thus ruling out solving the problem for a fixed $α$. Second, towards computing $α$-MMS+PO allocation for the best possible $α$, we obtain a dichotomous result: We derive two conditions and show that the problem is tractable under these two conditions, while dropping either renders the problem intractable. The two conditions are: (i) number of agents is a constant, and (ii) for every agent, her absolute value for all the items is at least a constant factor of her total (absolute) value for all the goods or all the chores. In particular, first, for instances satisfying (i) and (ii) we design a PTAS - an efficient algorithm to find an $(α-ε)$-MMS and $γ$-PO allocation when given $ε,γ>0$, for the highest possible $α\in(0,1]$. Second, we show that if either condition is not satisfied then finding an $α$-MMS allocation for any $α\in(0,1]$ is NP-hard, even when a solution exists for $α=1$. To the best of our knowledge, ours is the first algorithm that ensures both approximate MMS and PO guarantees.

cs.GT↗