论商品配置的最大公平性
On Allocating Goods to Maximize Fairness
摘要：
给定一组$m$代理和一组$n$项目，其中代理$A$具有项目$i$的效用$u_{A,i}$，我们的目标是将项目分配给代理以最大化公平性。具体来说，一个代理的效用是它收到的物品的效用之和，我们寻求最大化任何代理的最小效用。虽然这个问题最近受到了很多关注，但到目前为止，它的可近似性还没有得到很好的理解：最著名的近似算法实现了$\tilde{O}(\sqrt{m})$近似，相反，最著名的近似硬度为2。我们的主要结果是一个近似算法，实现了任何$\eps=Ω(\log\log n/\log n)$在时间$n^{O(1/\eps)}$的$\tilde{O}(n^{\eps})$近似。特别地，我们得到了拟多项式时间的多对数逼近，对于任意常数$\eps > 0$，我们得到了多项式时间的$O(n^{\eps})$逼近。我们的算法的一个有趣的方面是，我们使用作为一个积木的线性规划的完整性差距是$Ω(\sqrt m)$。我们通过迭代使用LP产生的解决方案来构建新的具有显著较小完整性间隙的实例，最终获得所需的近似值，从而绕过了这个障碍。我们还调查了特殊情况下的问题，每个项目有一个非零的效用最多两个代理。我们表明，即使在这种限制设置的问题是很难近似到任何因素比2，并显示一个因素$(2+\eps)$ -近似算法运行在时间$poly(n,1/\eps)$任何$\eps>0$。这种特殊的情况下，可以铸造作为一个图的边缘方向问题，我们的算法可以被看作是一个推广的欧拉方向加权图。
英文摘要：
Given a set of $m$ agents and a set of $n$ items, where agent $A$ has utility $u_{A,i}$ for item $i$, our goal is to allocate items to agents to maximize fairness. Specifically, the utility of an agent is the sum of its utilities for items it receives, and we seek to maximize the minimum utility of any agent. While this problem has received much attention recently, its approximability has not been well-understood thus far: the best known approximation algorithm achieves an $\tilde{O}(\sqrt{m})$-approximation, and in contrast, the best known hardness of approximation stands at 2. Our main result is an approximation algorithm that achieves an $\tilde{O}(n^{\eps})$ approximation for any $\eps=Ω(\log\log n/\log n)$ in time $n^{O(1/\eps)}$. In particular, we obtain poly-logarithmic approximation in quasi-polynomial time, and for any constant $\eps > 0$, we obtain $O(n^{\eps})$ approximation in polynomial time. An interesting aspect of our algorithm is that we use as a building block a linear program whose integrality gap is $Ω(\sqrt m)$. We bypass this obstacle by iteratively using the solutions produced by the LP to construct new instances with significantly smaller integrality gaps, eventually obtaining the desired approximation. We also investigate the special case of the problem, where every item has a non-zero utility for at most two agents. We show that even in this restricted setting the problem is hard to approximate upto any factor better tha 2, and show a factor $(2+\eps)$-approximation algorithm running in time $poly(n,1/\eps)$ for any $\eps>0$. This special case can be cast as a graph edge orientation problem, and our algorithm can be viewed as a generalization of Eulerian orientations to weighted graphs.
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