Testing Linear Properties: Some general themes

The last two decades have seen enormous progress in the development of sublinear-time algorithms — i.e., algorithms that examine/reveal properties of “data” in less time than it would take to read all of the data. A large, and important, subclass of such properties turn out to be “linear”. In particular, these developments have contributed to the rich theory of probabilistically checkable proofs (PCPs) and locally testable codes (LTCs). In this survey, we focus on some of the general technical themes at work behind the many results in this area.

[1]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[2]  Daniel A. Spielman,et al.  Nearly-linear size holographic proofs , 1994, STOC '94.

[3]  Oded Goldreich,et al.  Proximity Oblivious Testing and the Role of Invariances , 2011, APPROX-RANDOM.

[4]  Avi Wigderson,et al.  Symmetric LDPC codes and local testing , 2016, Comb..

[5]  Elena Grigorescu.  Symmetries in algebraic Property Testing , 2010 .

[6]  Madhu Sudan,et al.  Sparse Random Linear Codes are Locally Decodable and Testable , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[7]  Ronitt Rubinfeld,et al.  Sublinear Time Algorithms , 2011, SIAM J. Discret. Math..

[8]  Eli Ben-Sasson,et al.  Tensor Products of Weakly Smooth Codes are Robust , 2008, Theory Comput..

[9]  Madhu Sudan,et al.  2-Transitivity Is Insufficient for Local Testability , 2008, Computational Complexity Conference.

[10]  Shubhangi Saraf,et al.  Local list-decoding and testing of random linear codes from high error , 2010, STOC '10.

[11]  Shubhangi Saraf,et al.  Tolerant Linearity Testing and Locally Testable Codes , 2009, APPROX-RANDOM.

[12]  Madhu Sudan,et al.  Algebraic property testing: the role of invariance , 2008, Electron. Colloquium Comput. Complex..

[13]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[14]  Dana Ron,et al.  On proximity oblivious testing , 2009, STOC '09.

[15]  Oded Goldreich,et al.  Locally testable codes and PCPs of almost-linear length , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[16]  Eli Ben-Sasson,et al.  Symmetric LDPC Codes are not Necessarily Locally Testable , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[17]  Testing low-degree polynomials over prime fields , 2009 .

[18]  Eli Ben-Sasson,et al.  On Sums of Locally Testable Affine Invariant Properties , 2011, APPROX-RANDOM.

[19]  Eli Ben-Sasson,et al.  Short PCPs with Polylog Query Complexity , 2008, SIAM J. Comput..

[20]  Avi Wigderson,et al.  Robust local testability of tensor products of LDPC codes ? , 2006 .

[21]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[22]  Mihir Bellare,et al.  Linearity testing in characteristic two , 1996, IEEE Trans. Inf. Theory.

[23]  Eli Ben-Sasson,et al.  Composition of semi-LTCs by two-wise tensor products , 2009, computational complexity.

[24]  Shachar Lovett,et al.  Testing of exponentially large codes, by a new extension to Weil bound for character sums , 2010, Electron. Colloquium Comput. Complex..

[25]  Robust locally testable codes and products of codes , 2006 .

[26]  Venkatesan Guruswami,et al.  Locally Testable Codes Require Redundant Testers , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[27]  Madhu Sudan.  Invariance in Property Testing , 2010, Electron. Colloquium Comput. Complex..

[28]  Klaus Jansen,et al.  Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 12th International Workshop, APPROX 2009, and 13th International Workshop, RANDOM 2009, Berkeley, CA, USA, August 21-23, 2009. Proceedings , 2009, APPROX-RANDOM.

[29]  Eli Ben-Sasson,et al.  Composition of Semi-LTCs by Two-Wise Tensor Products , 2009, APPROX-RANDOM.

[30]  Madhu Sudan,et al.  Succinct Representation of Codes with Applications to Testing , 2009, SIAM J. Discret. Math..

[31]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[32]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[33]  Oded Goldreich,et al.  On The Randomness Complexity of Property Testing , 2009, computational complexity.

[34]  Eli Ben-Sasson,et al.  Low Rate Is Insufficient for Local Testability , 2010, APPROX-RANDOM.

[35]  Venkatesan Guruswami,et al.  Locally Testable Codes Require Redundant Testers , 2009, Computational Complexity Conference.

[36]  Oded Goldreich,et al.  Property Testing - Current Research and Surveys , 2010, Property Testing.

[37]  Noga Alon,et al.  Testing Reed-Muller codes , 2005, IEEE Transactions on Information Theory.

[38]  László Babai,et al.  Locally testable cyclic codes , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[39]  Dana Ron,et al.  Testing polynomials over general fields , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[40]  Atri Rudra,et al.  Testing low-degree polynomials over prime fields , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[41]  Simon Litsyn,et al.  Almost orthogonal linear codes are locally testable , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[42]  Or Meir,et al.  Combinatorial construction of locally testable codes , 2008, SIAM J. Comput..

[43]  Dana Ron,et al.  Algorithmic and Analysis Techniques in Property Testing , 2010, Found. Trends Theor. Comput. Sci..

[44]  Paul Valiant,et al.  The Tensor Product of Two Codes Is Not Necessarily Robustly Testable , 2005, APPROX-RANDOM.

[45]  Eli Ben-Sasson,et al.  Some 3CNF properties are hard to test , 2003, STOC '03.

[46]  Eli Ben-Sasson,et al.  Limits on the Rate of Locally Testable Affine-Invariant Codes , 2011, APPROX-RANDOM.

[47]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.