在线代理到达的杂务 MMS 分配

中文摘要 AI 辅助

我们研究了将 $m$ 个不可分割的杂务公平分配给 $n$ 个代理的问题，这些代理具有次加性成本函数，并以任意顺序在线到达。当代理到达时，我们会得知其成本函数，并且必须不可撤销地为其分配一组杂务。我们专注于最大最小份额（MMS）公平性概念，旨在计算一种分配，使得所有物品都被分配，并且没有代理承担超过其 MMS 的 $\alpha$ 倍的成本。在没有任何关于实例的先验信息（除了 $n$ 和 $m$）的情况下，我们设计了一种算法，其竞争比为 $O(\min\{n, k\log^{1+\epsilon}k, \log m\})$，适用于任意常数 $\epsilon > 0$，其中 $k$ 表示成本函数类型的数量。我们的界限与次加性成本下 MMS 的最佳已知离线近似保证相匹配，并且相对于所有三个参数而言几乎是最优的：我们表明，即使对于二元加性成本函数，也没有在线算法能够实现 $o(\min\{n, k\log k, \log m\})$ 的竞争比。然后，我们考虑 $k$ 种成本函数类型预先已知（尽管到达代理的实际类型未知）的设置。对于加性成本函数，我们提供了一种竞争比为 $O(\min\{\log k, \log(kn)/\log\log(kn)\})$ 的算法，并表明对于一般的 $k$，即使是在二元加性设置中，也不存在常数竞争比的算法。对于 $k \le n$ 的二元加性函数，我们提出了一种 $3$-竞争算法，并建立了 $2$ 的下界。

英文摘要

We study the fair allocation of $m$ indivisible chores to $n$ agents with subadditive cost functions arriving online in an arbitrary order. Upon an agent's arrival, we are informed of her cost function and must irrevocably assign her a set of chores. We focus on the Maximin Share (MMS) fairness notion and aim to compute an allocation in which all items are assigned, and no agent incurs a cost more than $α$ times her MMS. Without any prior information about the instance (other than $n$ and $m$), we design an algorithm with a competitive ratio of $O(\min\{n, k\log^{1+ε}k, \log m\})$ for any constant $ε> 0$, where $k$ denotes the number of cost function types. Our bound matches the best known offline approximation guarantees for MMS under subadditive costs and is nearly optimal with respect to all three parameters: we show that even for binary additive cost functions, no online algorithm can achieve a competitive ratio of $o(\min\{n, k\log k, \log m\})$. We then consider the setting in which the $k$ cost function types are known in advance (though the realized types of arriving agents are not). For additive cost functions, we provide an algorithm with a competitive ratio of $O(\min\{\log k, \log(kn)/\log\log(kn)\})$, and show that constant-competitive algorithms do not exist for general $k$, even for the binary additive setting. For binary additive functions when $k \le n$, we propose a $3$-competitive algorithm and establish a lower bound of $2$.

发表机构

• University of Macau（澳门大学）
• Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）

机构由 AI 辅助整理，请以论文原文为准。