Linear consistency testing

Yonatan Aumann; Johan Håstad; Michael O. Rabin; Madhu Sudany

doi:10.1007/978-3-540-48413-4_11

Linear consistency testing

Yonatan Aumann, Johan Håstad, Michael O. Rabin, Madhu Sudany

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

Abstract

We extend the notion of linearity testing to the task of checking linear-consistency of multiple functions. Informally, functions are “linear" if their graphs form straight lines on the plane. Two such functions are “consistent" if the lines have the same slope. We propose a variant of a test of Blum, Luby and Rubinfeld [8] to check the linear- consistency of three functions f₁; f₂; f₃ mapping a finite Abelian group G to an Abelian group H: Pick x; y ϵ G uniformly and independently at random and check if f₁(x) + f₂(y) = f₃(x + y). We analyze this test for two cases: (1) G and H are arbitrary Abelian groups and (2) G = {formula presened} and H = F₂. Questions bearing close relationship to linear-consistency testing seem to have been implicitly considered in recent work on the construction of PCPs (and in particular in the work of Håstad [9]). It is abstracted explicitly for the first time here. We give an application of this problem (and of our results): A (yet another) new and tight characterization of NP, namely 8∀ ϵ> 0; NP = MIP{formula presnted}. I.e., every language in NP has 3-prover 1-round proof systems in which the verifier tosses O(log n) coins and asks each of the three provers one question each. The provers respond with one bit each such that the verifier accepts instance of the language with probability 1–ϵ and rejects non-instances with probability at least {formula presented}. Such a result is of some interest in the study of probabilistically checkable proofs.

Original language	English
Title of host publication	Randomization, Approximation, and Combinatorial Optimization
Subtitle of host publication	Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings
Editors	Jose D. P. Rolim, Alistair Sinclair, Dorit Hochbaum, Klaus Jansen
Publisher	Springer Verlag
Pages	109-120
Number of pages	12
ISBN (Print)	3540663290, 9783540663294
DOIs	https://doi.org/10.1007/978-3-540-48413-4_11
State	Published - 1999
Event	3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999 - Berkeley, United States Duration: 8 Aug 1999 → 11 Aug 1999

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	1671
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999
Country/Territory	United States
City	Berkeley
Period	8/08/99 → 11/08/99

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1999.

Funding

We thank the reviewers of RANDOM’99 as well as the referee of the current paper for numerous comments and corrections. M.O.R.’s research was supported in part by NSF grant NSF-CCR-97-00365. M.S.’s research was supported in part by a Sloan Foundation Fellowship, and MIT-NEC Research Initiation Grant, and NSF Career Award CCR-9875511.

Funders	Funder number
MIT-NEC	CCR-9875511
Alfred P. Sloan Foundation

Access to Document

10.1007/978-3-540-48413-4_11

Cite this

Aumann, Y., Håstad, J., Rabin, M. O., & Sudany, M. (1999). Linear consistency testing. In J. D. P. Rolim, A. Sinclair, D. Hochbaum, & K. Jansen (Eds.), Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings (pp. 109-120). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1671). Springer Verlag. https://doi.org/10.1007/978-3-540-48413-4_11

Aumann, Yonatan ; Håstad, Johan ; Rabin, Michael O. et al. / Linear consistency testing. Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings. editor / Jose D. P. Rolim ; Alistair Sinclair ; Dorit Hochbaum ; Klaus Jansen. Springer Verlag, 1999. pp. 109-120 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "We extend the notion of linearity testing to the task of checking linear-consistency of multiple functions. Informally, functions are “linear{"} if their graphs form straight lines on the plane. Two such functions are “consistent{"} if the lines have the same slope. We propose a variant of a test of Blum, Luby and Rubinfeld [8] to check the linear- consistency of three functions f1; f2; f3 mapping a finite Abelian group G to an Abelian group H: Pick x; y ϵ G uniformly and independently at random and check if f1(x) + f2(y) = f3(x + y). We analyze this test for two cases: (1) G and H are arbitrary Abelian groups and (2) G = {formula presened} and H = F2. Questions bearing close relationship to linear-consistency testing seem to have been implicitly considered in recent work on the construction of PCPs (and in particular in the work of H{\aa}stad [9]). It is abstracted explicitly for the first time here. We give an application of this problem (and of our results): A (yet another) new and tight characterization of NP, namely 8∀ ϵ> 0; NP = MIP{formula presnted}. I.e., every language in NP has 3-prover 1-round proof systems in which the verifier tosses O(log n) coins and asks each of the three provers one question each. The provers respond with one bit each such that the verifier accepts instance of the language with probability 1–ϵ and rejects non-instances with probability at least {formula presented}. Such a result is of some interest in the study of probabilistically checkable proofs.",

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Aumann, Y, Håstad, J, Rabin, MO & Sudany, M 1999, Linear consistency testing. in JDP Rolim, A Sinclair, D Hochbaum & K Jansen (eds), Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1671, Springer Verlag, pp. 109-120, 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Berkeley, United States, 8/08/99. https://doi.org/10.1007/978-3-540-48413-4_11

Linear consistency testing. / Aumann, Yonatan; Håstad, Johan; Rabin, Michael O. et al.
Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings. ed. / Jose D. P. Rolim; Alistair Sinclair; Dorit Hochbaum; Klaus Jansen. Springer Verlag, 1999. p. 109-120 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1671).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Aumann Y, Håstad J, Rabin MO, Sudany M. Linear consistency testing. In Rolim JDP, Sinclair A, Hochbaum D, Jansen K, editors, Randomization, Approximation, and Combinatorial Optimization: Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings. Springer Verlag. 1999. p. 109-120. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-540-48413-4_11

Linear consistency testing

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