Improved constant approximation factor algorithms for $k$-center problem for uncertain data

TL;DR

This paper presents improved approximation algorithms for the $k$-center problem with uncertain data, achieving better approximation factors and extending solutions to restricted center sets in general metric spaces.

Contribution

It introduces a greedy algorithm with a 6-approximation for the uncertain $k$-center problem and improves the unrestricted approximation factor to 4 by considering finite center sets.

Findings

Improved the approximation factor from 10 to 6 for the unrestricted problem.

Achieved a 2-approximation for the restricted center set problem.

Enhanced the overall approximation factor to 4 with increased running time.

Abstract

In real applications, database systems should be able to manage and process data with uncertainty. Any real dataset may have missing or rounded values, also the values of data may change by time. So, it becomes important to handle these uncertain data. An important problem in database technology is to cluster these uncertain data. In this paper, we study the $k$-center problem for uncertain points in a general metric space. First we present a greedy approximation algorithm that builds $k$ centers using a farthest-first traversal in $k$ iterations. This algorithm improves the approximation factor of the unrestricted assigned $k$-center problem from $10$ to $6$. Next we restrict the centers to be selected from a finite set of points and we show that the optimal solution for this restricted setting is a $2$-approximation factor solution for the optimal solution of the assigned $k$-center problem. Using this idea we improve the approximation factor of the unrestricted assigned $k$-center problem to $4$ by increasing the running time mildly.