k-median: exact recovery in the extended stochastic ball model

• 成果类型: Article
• 署名作者: Del Pia, Alberto; Ma, Mingchen
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01886-5
• 发表日期: 2023
• 页码: 357-423
• 关键词: semidefinite; algorithm

摘要

We study exact recovery conditions for the linear programming relaxation of the k-median problem in the stochastic ball model (SBM). In Awasthi et al. (Relax, no need to round: integrality of clustering formulations. arXiv:1408.4045, 2015; in: Proceedings of the 2015 conference on innovations in theoretical computer science, pp 191-200, 2015), the authors give a tight result for the k-median LP in the SBM, saying that exact recovery can be achieved as long as the balls are pairwise disjoint. We give a counterexample to their result, thereby showing that the k-median LP is not tight in low dimension. Instead, we give a near optimal result showing that the k-median LP in the SBM is tight in high dimension. We also show that, if the probability measure satisfies some concentration assumptions, then the k-median LP in the SBM is tight in every dimension. Furthermore, we propose a new model of data called extended stochastic ball model (ESBM), which significantly generalizes the well-known SBM. We then show that exact recovery can still be achieved in the ESBM.