Outliers Detection Is Not So Hard: Approximation Algorithms for Robust Clustering Problems Using Local Search Techniques

TL;DR

This paper introduces new approximation algorithms for robust clustering problems like k-median and k-means with outliers, significantly improving their approximation ratios using local search techniques.

Contribution

It develops a novel analysis method for local search algorithms, achieving the best known approximation ratios for various robust clustering models.

Findings

Improved approximation ratio for k-Means with penalties from 25+ε to 9+ε.

Best known approximation ratio for k-Median with penalties.

Enhanced bi-criteria approximation algorithms for outlier models with tighter bounds.

Abstract

In this paper, we consider two types of robust models of the $k$-median/$k$-means problems: the outlier-version ($k$-MedO/$k$-MeaO) and the penalty-version ($k$-MedP/$k$-MeaP), in which we can mark some points as outliers and discard them. In $k$-MedO/$k$-MeaO, the number of outliers is bounded by a given integer. In $k$-MedP/$k$-MeaP, we do not bound the number of outliers, but each outlier will incur a penalty cost. We develop a new technique to analyze the approximation ratio of local search algorithms for these two problems by introducing an adapted cluster that can capture useful information about outliers in the local and the global optimal solution. For $k$-MeaP, we improve the best known approximation ratio based on local search from $25+\varepsilon$ to $9+\varepsilon$. For $k$-MedP, we obtain the best known approximation ratio. For $k$-MedO/$k$-MeaO, there exists only two bi-criteria approximation algorithms based on local search. One violates the outlier constraint (the constraint on the number of outliers), while the other violates the cardinality constraint (the constraint on the number of clusters). We consider the former algorithm and improve its approximation ratios from $17+\varepsilon$ to $3+\varepsilon$ for $k$-MedO, and from $274+\varepsilon$ to $9+\varepsilon$ for $k$-MeaO.