A Simple Combinatorial Algorithm for Robust Matroid Center

TL;DR

This paper introduces a simple greedy algorithm for the Robust Matroid Center problem, achieving a 5-approximation, improving simplicity and efficiency over previous LP-based methods.

Contribution

It presents a novel, straightforward greedy approach utilizing Rado matroids to approximate the Robust Matroid Center problem.

Findings

Achieves a 5-approximation with a simple greedy algorithm.

Uses Rado matroids to simplify the problem structure.

Improves upon previous LP-based approximation ratios.

Abstract

Recent progress on robust clustering led to constant-factor approximations for Robust Matroid Center. After a first combinatorial $7$-approximation that is based on a matroid intersection approach, two tight LP-based $3$-approximations were discovered, both relying on the Ellipsoid Method. In this paper, we show how a carefully designed, yet very simple, greedy selection algorithm gives a $5$-approximation. An important ingredient of our approach is a well-chosen use of Rado matroids. This enables us to capture with a single matroid a relaxed version of the original matroid, which, as we show, is amenable to straightforward greedy selections.