Private approximation of clustering and vertex cover

Amos Beimel, Renen Hallak, Kobbi Nissim

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

7 Scopus citations

Abstract

Private approximation of search problems deals with finding approximate solutions to search problems while disclosing as little information as possible. The focus of this work is on private approximation of the vertex cover problem and two well studied clustering problems -k-center and k-median. Vertex cover was considered in [Beimel, Carmi, Nissim, and Weinreb, STOC, 2006] and we improve their infeasibility results. Clustering algorithms are frequently applied to sensitive data, and hence are of interest in the contexts of secure computation and private approximation. We show that these problems do not admit private approximations, or even approximation algorithms that leak significant number of bits. For the vertex cover problem we show a tight infeasibility result: every algorithm that p(n)-approximates vertex-cover must leak Ω(n/ρ(n)) bits (where n is the number of vertices in the graph). For the clustering problems we prove that even approximation algorithms with a poor approximation ratio must leak Ω(n) bits (where n is the number of points in the instance). For these results we develop new proof techniques, which are more simple and intuitive than those in Beimel et al., and yet allow stronger infeasibility results. Our proofs rely on the hardness of the promise problem where a unique optimal solution exists [Valiant and Vazirani, Theoretical Computer Science, 1986], on the hardness of approximating witnesses for NP-hard problems ([Kumar and Sivakumar, CCC, 1999] and [Feige, Langberg, and Nissim, APPROX, 2000]), and on a simple random embedding of instances into bigger instances.

Original languageEnglish
Title of host publicationTheory of Cryptography - 4th Theory of Cryptography Conference, TCC 2007, Proceedings
PublisherSpringer Verlag
Pages383-403
Number of pages21
ISBN (Print)9783540709350
DOIs
StatePublished - 1 Jan 2007
Event4th Theory of Cryptography Conference, TCC 2OO7 - Amsterdam, Netherlands
Duration: 21 Feb 200724 Feb 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4392 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference4th Theory of Cryptography Conference, TCC 2OO7
Country/TerritoryNetherlands
CityAmsterdam
Period21/02/0724/02/07

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