Coreset-based Strategies for Robust Center-type Problems

TL;DR

This paper introduces coreset-based algorithms for robust center problems, achieving near-optimal approximation ratios with efficient, scalable solutions adaptable to dataset complexity.

Contribution

It presents novel coreset strategies for robust matroid and knapsack center problems, enabling efficient algorithms with near-optimal approximation ratios and adaptability to dataset complexity.

Findings

Achieves (3+ε)-approximation for robust center problems.

Provides algorithms with linear time complexity in dataset size.

Develops scalable MapReduce and Streaming algorithms with few rounds/passes.

Abstract

Given a dataset $V$ of points from some metric space, the popular $k$-center problem requires to identify a subset of $k$ points (centers) in $V$ minimizing the maximum distance of any point of $V$ from its closest center. The \emph{robust} formulation of the problem features a further parameter $z$ and allows up to $z$ points of $V$ (outliers) to be disregarded when computing the maximum distance from the centers. In this paper, we focus on two important constrained variants of the robust $k$-center problem, namely, the Robust Matroid Center (RMC) problem, where the set of returned centers are constrained to be an independent set of a matroid of rank $k$ built on $V$, and the Robust Knapsack Center (RKC) problem, where each element $i\in V$ is given a positive weight $w_i<1$ and the aggregate weight of the returned centers must be at most 1. We devise coreset-based strategies for the two problems which yield efficient sequential, MapReduce, and Streaming algorithms. More specifically, for any fixed $\epsilon>0$, the algorithms return solutions featuring a $(3+\epsilon)$-approximation ratio, which is a mere additive term $\epsilon$ away from the 3-approximations achievable by the best known polynomial-time sequential algorithms for the two problems. Moreover, the algorithms obliviously adapt to the intrinsic complexity of the dataset, captured by its doubling dimension $D$. For wide ranges of the parameters $k,z,\epsilon, D$, we obtain a sequential algorithm with running time linear in $|V|$, and MapReduce/Streaming algorithms with few rounds/passes and substantially sublinear local/working memory.