Improvements on approximation algorithms for clustering probabilistic data

摘要

Uncertainty about data appears in many real-world applications and an important issue is how to manage, analyze and solve optimization problems over such data. An important tool for data analysis is clustering. When the data set is uncertain, we can model them as a set of probabilistic points each formalized as a probability distribution function which describes the possible locations of the points. In this paper, we study k-center problem for probabilistic points in a general metric space. First, we present a fast greedy approximation algorithm that builds k centers using a farthest-first traversal in k iterations. This algorithm improves the previous approximation factor of the unrestricted assigned k-center problem from 10 (see [1]) to 6. Next, we restrict the centers to be selected from all the probabilistic locations of the given points and we show that an optimal solution for this restricted setting is a 2-approximation factor solution for an optimal solution of the assigned k-center problem with expected distance assignment. Using this idea, we improve the approximation factor of the unrestricted assigned k-center problem to 4 by increasing the running time. The algorithm also runs in polynomial time when k is a constant. Additionally, we implement our algorithms on three real data sets. The experimental results show that in practice the approximation factors of our algorithms are better than in theory for these data sets. Also we compare the results of our algorithm with the previous works and discuss about the achieved results. At the end, we present our theoretical results for probabilistic k-median clustering.