Article

Private approximation of NP-hard functions

Authors:

Shai Halevi,

Robert Krauthgamer,

Eyal Kushilevitz,

Kobbi NissimAuthors Info & Claims

STOC '01: Proceedings of the thirty-third annual ACM symposium on Theory of computing

Pages 550 - 559

https://doi.org/10.1145/380752.380850

Published: 06 July 2001 Publication History

Get Access

Abstract

The notion of private approximation was introduced recently by Feigenbaum, Fong, Strauss and Wright. Informally, a private approximation of a function f is another function F that approximates f in the usual sense, but does not yield any information on x other than what can be deduced from f(x). As such, F(x) is useful for private computation of f(x) (assuming that F can be computed more efficiently than f.

In this work we examine the properties and limitations of this new notion. Specifically, we show that for many NP-hard problems, the privacy requirement precludes non-trivial approximation. This is the case even for problems that otherwise admit very good approximation (e.g., problems with PTAS). On the other hand, we show that slightly relaxing the privacy requirement, by means of leaking “just a few bits of informationrdquo; about x, again permits good approximation.

References

[1]

T. Asano and D. P. Williamson. Improved approximation algorithms for MAX SAT. In 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 96-105. ACM, New York, 2000.]]

Digital Library

Google Scholar

[2]

G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. Complexity and Approximation: Combinatorial optimization problems and their approximability properties. Springer Verlag, 1999. Includes a compendium of NP optimization problems, which is also available at http://www.nada.kth.se/~viggo/wwwcompendium/.]]

Digital Library

Google Scholar

[3]

B. S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41(1):153-180, 1994.]]

Digital Library

Google Scholar

[4]

R. Bar-Yehuda, B. Chor, E. Kushilevitz, and A. Orlitsky. Privacy, additional information, and communication. IEEE Transactions on Information Theory, 39(6):1930-1943, 1993.]]

Digital Library

Google Scholar

[5]

U. Feige, S. Goldwasser, L. Lovasz, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268-292, 1996.]]

Digital Library

Google Scholar

[6]

J. Feigenbaum, J. Fong, M. Strauss, and R. N. Wright. Secure multiparty computation of approximations. Unpublished manuscript. Presented at DIMACS Workshop on Cryptography and Intractability. March 2000.]]

Google Scholar

[7]

J. Feigenbaum, Y. Ishai, T. Malkin, K. Nissim, M. Strauss, and R. N. Wright. Secure multiparty computation of approximations. To appear in ICALP 2001. A longer version is available as an eprint report http://eprint.iacr.org/2001/024.]]

Digital Library

Google Scholar

[8]

M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H.Freeman & Co., 1979.]]

Digital Library

Google Scholar

[9]

O. Goldreich. Foundations of cryptography (fragments of a book). 1995. available as a monograph from the Electronic Colloquium on Computational Complexity, http://www.eccc.uni-trier.de/eccc/.]]

Google Scholar

[10]

O. Goldreich, S. Micali, and A. Wigderson. Proofs that yield nothing but their validity and a methodology of cryptographic protocol design. In 27th Annual Symposium on Foundations of Computer Science, pages 174-187. IEEE, 1986.]]

Digital Library

Google Scholar

[11]

O. Goldreich and E. Petrank. Quantifying knowledge complexity. Computational Complexity, 8(1):50-98, 1999. Preliminary version appeared in FOCS'91, pp. 59-68.]]

Digital Library

Google Scholar

[12]

J. Hastad. Some optimal inapproximability results. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 1-10. ACM, 1997. Also available as Report TR97-037 from ECCC, http://www.eccc.uni-trier.de/eccc/.]]

Digital Library

Google Scholar

[13]

D. S. Hochbaum, editor. Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston, MA, 1997.]]

Digital Library

Google Scholar

Cited By

View all

Choi SDachman-soled DKulkarni MYerukhimovich A(2020)Differentially-Private Multi-Party Sketching for Large-Scale StatisticsProceedings on Privacy Enhancing Technologies10.2478/popets-2020-00472020:3(153-174)Online publication date: 17-Aug-2020
https://doi.org/10.2478/popets-2020-0047
Berger BBrakerski Z(2018)Zero-Knowledge Protocols for Search ProblemsSecurity and Cryptography for Networks10.1007/978-3-319-98113-0_16(292-309)Online publication date: 3-Aug-2018
https://doi.org/10.1007/978-3-319-98113-0_16
Meng XKamara SNissim KKollios GRay ILi NKruegel C(2015)GRECSProceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security10.1145/2810103.2813672(504-517)Online publication date: 12-Oct-2015
https://dl.acm.org/doi/10.1145/2810103.2813672
Show More Cited By

Index Terms

Private approximation of NP-hard functions
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Recommendations

NP-Hard Sets Are Exponentially Dense Unless coNP C NP/poly
CCC '08: Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity

We show that hard sets $S$ for $\NP$ must have exponential density, i.e. $|S_{=n}| \geq 2^{n^\epsilon}$ for some $\epsilon > 0$ and infinitely many $n$, unless $\coNP \subseteq \NP/\poly$ and the polynomial-time hierarchy collapses. This result holds ...
Which NP-Hard Optimization Problems Admit Non-trivial Efficient Approximation Algorithms?
ICALP '00: Proceedings of the 27th International Colloquium on Automata, Languages and Programming

In 1991 Feige, Goldwasser, Lovász, Safra and Szegedy found a connection between the approximability of Max-Clique and the area of multiprover interactive proofs. What first seemed like an isolated result have developed into an entire area of research.

...
Closeness of NP-Hard Sets to Other Complexity Classes

Let $A$ be a language and $C$ be a class of languages. $A$ is said to be s-C-close (s-C-outside-close) if there exists $B \in C$ such that $\parallel (A \bigtriangleup B)^{\leq n}\parallel \leq s(n)$ (and $A \subseteq B $). If $A$ is q-C-close (q-C-...

Comments

Information & Contributors

Information

Published In

STOC '01: Proceedings of the thirty-third annual ACM symposium on Theory of computing

July 2001

755 pages

ISBN:1581133499

DOI:10.1145/380752

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 06 July 2001

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

STOC01

Sponsor:

SIGACT

STOC01: 33rd ACM Symposium on Theory of Computing

Hersonissos, Greece

Acceptance Rates

STOC '01 Paper Acceptance Rate 83 of 230 submissions, 36%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

23
Total Citations
View Citations
41
Total Downloads

Downloads (Last 12 months)3
Downloads (Last 6 weeks)0

Reflects downloads up to 10 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Choi SDachman-soled DKulkarni MYerukhimovich A(2020)Differentially-Private Multi-Party Sketching for Large-Scale StatisticsProceedings on Privacy Enhancing Technologies10.2478/popets-2020-00472020:3(153-174)Online publication date: 17-Aug-2020
https://doi.org/10.2478/popets-2020-0047
Berger BBrakerski Z(2018)Zero-Knowledge Protocols for Search ProblemsSecurity and Cryptography for Networks10.1007/978-3-319-98113-0_16(292-309)Online publication date: 3-Aug-2018
https://doi.org/10.1007/978-3-319-98113-0_16
Meng XKamara SNissim KKollios GRay ILi NKruegel C(2015)GRECSProceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security10.1145/2810103.2813672(504-517)Online publication date: 12-Oct-2015
https://dl.acm.org/doi/10.1145/2810103.2813672
Feigenbaum JJaggard ASchapira M(2014)Approximate PrivacyACM Transactions on Algorithms10.1145/260106710:3(1-38)Online publication date: 1-Jul-2014
https://dl.acm.org/doi/10.1145/2601067
Boyle EGoldwasser STessaro S(2013)Communication locality in secure multi-party computationProceedings of the 10th theory of cryptography conference on Theory of Cryptography10.1007/978-3-642-36594-2_21(356-376)Online publication date: 3-Mar-2013
https://dl.acm.org/doi/10.1007/978-3-642-36594-2_21
Woodruff DFortnow LVadhan S(2011)Near-optimal private approximation protocols via a black box transformationProceedings of the forty-third annual ACM symposium on Theory of computing10.1145/1993636.1993733(735-744)Online publication date: 6-Jun-2011
https://dl.acm.org/doi/10.1145/1993636.1993733
Gupta ALigett KMcSherry FRoth ATalwar KCharikar M(2010)Differentially private combinatorial optimizationProceedings of the twenty-first annual ACM-SIAM symposium on Discrete algorithms10.5555/1873601.1873691(1106-1125)Online publication date: 17-Jan-2010
https://dl.acm.org/doi/10.5555/1873601.1873691
Feigenbaum JJaggard ASchapira MParkes DDellarocas CTennenholtz M(2010)Approximate privacyProceedings of the 11th ACM conference on Electronic commerce10.1145/1807342.1807369(167-178)Online publication date: 7-Jun-2010
https://dl.acm.org/doi/10.1145/1807342.1807369
Beimel AMalkin TNissim KWeinreb E(2010)How Should We Solve Search Problems Privately?Journal of Cryptology10.1007/s00145-008-9032-z23:2(344-371)Online publication date: 1-Apr-2010
https://dl.acm.org/doi/10.1007/s00145-008-9032-z
Gondree MMohassel PAl-Shaer EParaboschi S(2009)Longest common subsequence as private searchProceedings of the 8th ACM workshop on Privacy in the electronic society10.1145/1655188.1655200(81-90)Online publication date: 9-Nov-2009
https://dl.acm.org/doi/10.1145/1655188.1655200
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Cited By

Index Terms

Recommendations

NP-Hard Sets Are Exponentially Dense Unless coNP C NP/poly

Which NP-Hard Optimization Problems Admit Non-trivial Efficient Approximation Algorithms?

Closeness of NP-Hard Sets to Other Complexity Classes

Comments

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Conference

Acceptance Rates

Other Metrics

Article Metrics

Other Metrics

Cited By

Login options

Full Access

PDF

eReader

Abstract

References

Cited By

Index Terms

Recommendations

NP-Hard Sets Are Exponentially Dense Unless coNP C NP/poly

Which NP-Hard Optimization Problems Admit Non-trivial Efficient Approximation Algorithms?

Closeness of NP-Hard Sets to Other Complexity Classes

Comments

Information

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Conference

Acceptance Rates

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

Get Access

Login options

Full Access

View options

PDF

eReader

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations