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：针对具有离群点的容错$k$-供应商（$\mathsf{F}k\mathsf{SO}$）问题，本文提出了近似算法。该问题是两个已知问题（带离群点的$k$-供应商问题和容错$k$-供应商问题）的通用推广，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，目标是通过从可能的设施集合$F$中开放$k$个设施来服务$n$个客户$C$；其目标函数是任意客户到达开放设施所需的最远距离。在$\mathsf{F}k\mathsf{SO}$中，每个客户$v$具有容错需求$\ell_v$，即需要$\ell_v$个设施为其服务；因此每个客户$v$对目标函数的贡献是其到第$\ell_v^{\text{th}}$近开放设施的距离。此外，我们可选择$m$个要服务的客户，仅这些客户对目标函数有贡献，而其余$n-m$个客户被视为离群点。本文的主要结果为$\mathsf{F}k\mathsf{SO}$问题的一个$\min\{4t-1,2^t+1\}$近似算法，其中$t$为实例中出现的不同$\ell_v$值的数量。当$t=1$，即所有$\ell_v$统一为某个常数$\ell$时，该算法达到3-近似，优于Inamdar和Varadarajan[2020]针对均匀情形给出的11-近似（这两位学者也是该问题的提出者）。本文对均匀情形的结果匹配了$k$-供应商、带离群点的$k$-供应商以及容错$k$-供应商问题中存在的紧的3-近似。本文的核心技术贡献在于将round-or-cut框架应用于$\mathsf{F}k\mathsf{SO}$问题。在LP松弛的指导下，我们将问题简化为一个更简单的优化问题，通过求解该问题，可为"round"步骤获得距离界限，并为"cut"步骤获得有效不等式。