We present approximation algorithms for the Fault-tolerant $k$-Supplier with Outliers ($\mathsf{F}k\mathsf{SO}$) problem. This is a common generalization of two known problems -- $k$-Supplier with Outliers, and Fault-tolerant $k$-Supplier -- each of which generalize the well-known $k$-Supplier problem. In the $k$-Supplier problem the goal is to serve $n$ clients $C$, by opening $k$ facilities from a set of possible facilities $F$; the objective function is the farthest that any client must travel to access an open facility. In $\mathsf{F}k\mathsf{SO}$, each client $v$ has a fault-tolerance $\ell_v$, and now desires $\ell_v$ facilities to serve it; so each client $v$'s contribution to the objective function is now its distance to the $\ell_v^{\text{th}}$ closest open facility. Furthermore, we are allowed to choose $m$ clients that we will serve, and only those clients contribute to the objective function, while the remaining $n-m$ are considered outliers. Our main result is a $\min\{4t-1,2^t+1\}$-approximation for the $\mathsf{F}k\mathsf{SO}$ problem, where $t$ is the number of distinct values of $\ell_v$ that appear in the instance. At $t=1$, i.e. in the case where the $\ell_v$'s are uniformly some $\ell$, this yields a $3$-approximation, improving upon the $11$-approximation given for the uniform case by Inamdar and Varadarajan [2020], who also introduced the problem. Our result for the uniform case matches tight $3$-approximations that exist for $k$-Supplier, $k$-Supplier with Outliers, and Fault-tolerant $k$-Supplier. Our key technical contribution is an application of the round-or-cut schema to $\mathsf{F}k\mathsf{SO}$. Guided by an LP relaxation, we reduce to a simpler optimization problem, which we can solve to obtain distance bounds for the "round" step, and valid inequalities for the "cut" step.

翻译：我们针对带有离群点的容错 $k$-供应商问题（$\mathsf{F}k\mathsf{SO}$）提出了近似算法。该问题是两个已知问题——带有离群点的 $k$-供应商问题和容错 $k$-供应商问题——的共同推广，而这两个问题本身又推广了著名的 $k$-供应商问题。在 $k$-供应商问题中，目标是从一组候选设施 $F$ 中开设 $k$ 个设施来服务 $n$ 个客户 $C$；目标函数是任何客户前往开放式设施所需的最大距离。在 $\mathsf{F}k\mathsf{SO}$ 中，每个客户 $v$ 具有容错能力 $\ell_v$，并且需要 $\ell_v$ 个设施来服务它；因此每个客户 $v$ 对目标函数的贡献是其到第 $\ell_v$ 近的开放式设施的距离。此外，我们允许选择 $m$ 个客户进行服务，只有这些客户计入目标函数，而剩余的 $n-m$ 个客户被视为离群点。我们的主要结果是针对 $\mathsf{F}k\mathsf{SO}$ 问题的一个 $\min\{4t-1,2^t+1\}$-近似算法，其中 $t$ 是实例中出现的 $\ell_v$ 的不同值数量。在 $t=1$ 的情况下，即当 $\ell_v$ 统一为某个 $\ell$ 时，这给出了一个 $3$-近似，改进了 Inamdar 和 Varadarajan [2020] 针对均匀情况给出的 $11$-近似，后者也引入了该问题。我们针对均匀情况的结果与 $k$-供应商问题、带有离群点的 $k$-供应商问题以及容错 $k$-供应商问题中存在的紧的 $3$-近似相匹配。我们的关键技术贡献是将 round-or-cut 方案应用于 $\mathsf{F}k\mathsf{SO}$。在线性规划松弛的指导下，我们将其简化为一个更简单的优化问题，通过解决该问题，我们可以为“舍入”步骤获得距离界，并为“切割”步骤获得有效不等式。