Generalized Center Problems with Outliers

TL;DR

This paper introduces a dichotomy theorem for the generalized $ ext{F}$-center problem with outliers, providing a 3-approximation algorithm under certain conditions, and applies it to solve the knapsack center problem with outliers.

Contribution

It establishes a dichotomy theorem characterizing when a 3-approximation is possible for the problem, and offers the first polynomial-time 3-approximation for the knapsack center problem with outliers.

Findings

Efficient 3-approximation exists if and only if a certain linear optimization is feasible.

Polynomial time 3-approximation achieved for the knapsack center problem with outliers.

Provides a theoretical framework linking approximation algorithms to linear optimization over family $ ext{F}$.

Abstract

We study the $\mathcal{F}$-center problem with outliers: given a metric space $(X,d)$, a general down-closed family $\mathcal{F}$ of subsets of $X$, and a parameter $m$, we need to locate a subset $S\in \mathcal{F}$ of centers such that the maximum distance among the closest $m$ points in $X$ to $S$ is minimized. Our main result is a dichotomy theorem. Colloquially, we prove that there is an efficient $3$-approximation for the $\mathcal{F}$-center problem with outliers if and only if we can efficiently optimize a poly-bounded linear function over $\mathcal{F}$ subject to a partition constraint. One concrete upshot of our result is a polynomial time $3$-approximation for the knapsack center problem with outliers for which no (true) approximation algorithm was known.