Fault-tolerant $k$-Supplier with Outliers

TL;DR

The paper tackles Fault-tolerant $k$-Supplier with Outliers (FkSO), a generalization of clustering problems that incorporates nonuniform fault-tolerances and an outlier budget. It develops a strong LP relaxation and employs a round-or-cut framework to derive approximation algorithms that scale with the number of distinct fault-tolerance values $t$, achieving a $( ext{min}\{4t-1,2^t+1 ho brace})$-approximation, and a separate $(2^t+1)$-approximation pathway, with a 3-approximation in the uniform case UF$k$SO. The approach hinges on the notion of good partitions and a budgeting subroutine that distributes facilities across partition parts, using LP-guided rounding and dual-based cuts to either round to a feasible solution or derive a violated inequality. The results advance understanding of clustering with outliers under nonuniform fault-tolerance, while highlighting inherent limits (e.g., an $oldsymbol{ ext{Omega}}(t)$ barrier) and open questions about constant-factor approximations for general $t$.

Abstract

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.

Figures (3)

• Figure 1: One of the $k$ identical gadgets in the gap example, showing LP values in red ($x$ values) and blue ($\mathsf{cov}$ values). The "edges" represent distance $1$, and all other distances are determined by making triangle inequalities tight. The fault-tolerances are $\ell_{v_1} = \ell_{v_2} = \cdots = \ell_{v_k} = k$, and $\ell_v = 1$.
• Figure 2: An example of a $(6,\mathsf{cov})$-good partition $\mathcal{P}$ (\ref{['def:good-partition']}). The ellipses represent $\mathcal{P}$, and their subdivisions represent the $\mathsf{child}$ sets. All the circles are clients, with the filled-in circles being $R$, and among those, the double borders indicate the $j_P$'s. $\mathsf{cov}$ values are $1$ on $R$ and $1/2$ elsewhere. $\ell_v$ values are indicated by the sizes of the circles. The "edges" represent distance $2r$, and all other distances are obtained by making triangle inequalities tight.
• Figure 3: An example showing the limitations of good partitions, with a solution to \ref{['lp:fkso:m']}-\ref{['lp:fkso:bounds']} shown in red ($z$ values) and blue ($\mathsf{cov}$ values). The thin "edges" represent distance $1$, the thick "edges" represent distance $2$, and all other distances are determined by making triangle inequalities tight. The fault-tolerances are $\ell_{v_1} = 1, \ell_{v_2} = 2, \dots, \ell_{v_t} = t$.

Theorems & Definitions (30)

• Definition 1
• Definition 2: Fault-tolerant $k$-Supplier and Fault-tolerant $k$-Supplier with Outliers
• Definition 3: $r$-well-separated set
• Theorem 4
• Definition 5: Uniformly Fault-tolerant $k$-Supplier with Outliers ( UF$k$ SO)
• Theorem 6
• Claim 7
• Lemma 8
• proof
• Lemma 9