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

Rucha Kulkarni, R. Mehta, Setareh Taki
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引用次数: 15

Abstract

In this paper we study the problem of finding fair and efficient allocations of a mixed manna, i.e., a setM of discrete items that are goods/chores, among a set N of agents with additive valuations. We note that a mixed manna allows an item to be a good (positively valued) for some agents, and a chore (negatively valued) for others, and thereby strictly generalizes the extensively studied goods (chores) manna. To measure fairness and efficiencywe consider the popular and well studied notions of maximin-share (MMS) and Pareto optimality (PO) respectively. AnMMS allocation is one where every agent gets at least herMMS value. However, [6] showed that an MMS allocation may not always exist. This prompted a series of works on the efficient computation of α-MMS allocations, where every agent gets at least α (1/α) times her MMS value for a goods (chores) manna, for progressively better α ∈ [0, 1]; the best factor known so far is α = (3/4 + 1/(12n)) by Garg and Taki [2] for n ≥ 5 agents for goods, and 9/11 for chores [5]. No such results are known for the mixed manna. Even for the goods (chores) manna, no work has explored the PO guarantee in addition toMMS, to the best of our knowledge. In this paper, we first show that, for any fixed α ∈ (0, 1], an α-MMS allocation may not always exist; in contrast, non-existence with a goods manna is known for α close to one. This rules out efficient computation for any fixed α, and naturally raises the following problem. Problem of Interest. Design an efficient algorithm to find an α-MMS + PO allocation for the best possible α , i.e., the maximum α ∈ (0, 1] for which it exists. This exact problem is intractable: In the case of identical agents, an (α = 1)-MMS allocation exists by definition. However, finding one is known to be NP-hard for a goods manna. On the positive side, a polynomial-time approximation scheme (PTAS) is known for this case; given a constant ε ∈ (0, 1], the algorithm finds a (1 − ε)-MMS allocation in polynomial time. Guaranteeing PO in addition adds to the complexity, since even checking if a given allocation is PO is coNP-hard. In light of these results, we ask,
不可分的混合甘露:关于MMS+PO分配的可计算性
在本文中,我们研究了在一组N个具有可加性估值的代理中寻找公平有效分配混合甘纳的问题,即一组作为商品/杂务的离散项目。我们注意到,混合吗哪允许一种物品对某些代理人来说是好的(正面评价),而对另一些代理人来说是杂务(负面评价),因此严格概括了广泛研究的商品(杂务)吗哪。为了衡量公平和效率,我们分别考虑了最优份额(MMS)和帕累托最优(PO)这两个流行且研究得很好的概念。mms分配是指每个代理至少获得一个mms值。然而,[6]表明MMS分配可能并不总是存在。这促使了一系列关于α-MMS分配的有效计算的工作,其中每个代理至少获得α (1/α)倍于其商品(家务)吗哪的MMS值,对于逐渐更好的α∈[0,1];到目前为止,已知的最佳因子是Garg和Taki[2]对于n≥5个代理的商品的α = (3/4 + 1/(12n)),以及9/11对于家务的[5]。混合甘露没有这样的结果。据我们所知,即使对于货物(家务)吗哪,除了toMMS之外,还没有任何工作探索过PO保证。本文首先证明了对于任意固定的α∈(0,1),α- mms分配可能不总是存在;相反,不存在的好吗哪是已知的α接近1。这就排除了对任何固定α的有效计算,自然会产生以下问题。利益问题。设计一种有效的算法,寻找α- mms + PO分配的最佳可能α,即α∈(0,1]存在的最大值。这个问题很棘手:在相同代理的情况下,根据定义存在(α = 1)-MMS分配。然而,要找到一种甘露是很困难的。在积极的一面,多项式时间近似方案(PTAS)是已知的这种情况;给定常数ε∈(0,1),该算法在多项式时间内找到(1−ε)-MMS分配。另外,保证PO增加了复杂性,因为即使检查给定分配是否为PO也是cp -hard的。根据这些结果,我们问,
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