eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-13
9:1
9:16
10.4230/LIPIcs.APPROX-RANDOM.2018.9
article
Perturbation Resilient Clustering for k-Center and Related Problems via LP Relaxations
Chekuri, Chandra
1
Gupta, Shalmoli
1
Department of Computer Science, University of Illinois, Urbana-Champaign, IL 61801, USA
We consider clustering in the perturbation resilience model that has been studied since the work of Bilu and Linial [Yonatan Bilu and Nathan Linial, 2010] and Awasthi, Blum and Sheffet [Awasthi et al., 2012]. A clustering instance I is said to be alpha-perturbation resilient if the optimal solution does not change when the pairwise distances are modified by a factor of alpha and the perturbed distances satisfy the metric property - this is the metric perturbation resilience property introduced in [Angelidakis et al., 2017] and a weaker requirement than prior models. We make two high-level contributions.
- We show that the natural LP relaxation of k-center and asymmetric k-center is integral for 2-perturbation resilient instances. We belive that demonstrating the goodness of standard LP relaxations complements existing results [Maria{-}Florina Balcan et al., 2016; Angelidakis et al., 2017] that are based on new algorithms designed for the perturbation model.
- We define a simple new model of perturbation resilience for clustering with outliers. Using this model we show that the unified MST and dynamic programming based algorithm proposed in [Angelidakis et al., 2017] exactly solves the clustering with outliers problem for several common center based objectives (like k-center, k-means, k-median) when the instances is 2-perturbation resilient. We further show that a natural LP relxation is integral for 2-perturbation resilient instances of k-center with outliers.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol116-approx-random2018/LIPIcs.APPROX-RANDOM.2018.9/LIPIcs.APPROX-RANDOM.2018.9.pdf
Clustering
Perturbation Resilience
LP Integrality
Outliers
Beyond Worst Case Analysis