Approximation Algorithms for Robust Clustering Problems Using Local Search Techniques

In this paper, we explore two robust models for the k-median and k-means problems: the outlier-version (k-MedO/k-MeaO) and the penalty-version (k-MedP/k-MeaP), enabling the marking and elimination of certain points as outliers. In k-MedO/k-MeaO, the count of outliers is restricted by a specified integer, while in k-MedP/k-MeaP, there’s no explicit limit on outlier quantity, yet each outlier incurs a penalty cost. We introduce a novel approach to evaluate the approximation ratio of local search algorithms for these problems. This involves an adapted clustering method that captures pertinent information about outliers within both local and global optimal solutions. For k-MeaP, we enhance the best-known approximation ratio derived from local search, elevating it from \(25+\varepsilon \) to \(9+\varepsilon \) . The best-known approximation ratio for k-MedP is also obtained. Regarding k-MedO/k-MeaO, only two bi-criteria approximation algorithms based on local search exist. One violates the outlier constraint (limiting outlier count), while the other breaches the cardinality constraint (restricting the number of clusters). We focus on the former algorithm, enhancing its approximation ratios from \(17+\varepsilon \) to \(3+\varepsilon \) for k-MedO and from \(274+\varepsilon \) to \(9+\varepsilon \) for k-MeaO.