| 1 |
2016.08.24 |
Introduction introduction to the course general formulation of property testing [GGR98, introduction] testing linearity BLR test [BLR90] analysis via majority decoding [BLR90] testing regularity in the dense graph model via degree estimation [GGR98, Proposition 10.2.1.3 and Appendix D] via 2 queries (proximity oblivious testing) even two queries suffice to get information on regularity! [GS12, introduction and Section 3.1] |
Main: Blum Luby Rubinfeld 1990: Self-testing/correcting with applications to numerical problems Goldreich Goldwasser Ron 1998: Property testing and its connection to learning and approximation Additional: Goldreich Shinkar 2012: Two-sided error proximity oblivious testing |
| 2 |
2016.08.31 |
Boolean Functions 1 testing linearity over binary fields analysis via Fourier coefficients [BCHKS96, introduction and Section 2] 3delta lower bound [BGLR93, Lemma 3.7] testing linear consistency BLR test with three functions [AHRS99, Section 4] efficient soundness amplification analysis of graph tests [HW03, Section 3] |
Main Bellare Coppersmith Håstad Kiwi Sudan 1996: Linearity testing in characteristic two first connection of Fourier analysis and testing Aumann Håstad Rabin Sudan 1999: Linear consistency testing modification of BLR that sends each query to a different function Håstad Wigderson 2003: Simple analysis of graph tests for linearity and PCP efficient soundness amplification Additional: Bellare Goldwasser Lund Russel 1993: Efficient probabilistically checkable proofs and applications to approximations Kaufman Litsyn Xie 2007: Breaking the ε-soundness bound of the linearity test over GF(2) Kiwi 1996: Probabilistically checkable proofs and the testing of Hadamard-like codes improved analysis of linearity testing over all finite fields Samorodnitsky Trevisan 2005: Gowers uniformity, influence of variables, and PCPs |
| 3 |
2016.09.07 |
Boolean Functions 2 testing dictators definition of dictator 4-query test Håstad test [H01, Section 4] Rubinfeld's lecture notes testing juntas definition of k-junta [B09, Section 2] adaptive O(k*logk+k/ε)-query test [B09, Sections 3,4] |
Main: Håstad 2001: Some optimal inapproximability results Blais 2009: Testing juntas nearly optimally Additional on testing dictators and their applications: (video) Håstad 2013: Inapproximability of CSPs I (video) Håstad 2013: Inapproximability of CSPs II (video) Håstad 2013: Inapproximability of CSPs III (video) Håstad 2013: Inapproximability of CSPs IV (video) Håstad 2013: Inapproximability of CSPs V Additional on juntas: Fischer Kindler Ron Safra Samorodnitsky 2004: Testing juntas polynomial-query tester Servedio Tan Wright 2015: Adaptivity helps for testing juntas lower bound on non-adaptive testers for k-juntas Additional on PCP applications of Fourier analys: Khot Saket 2006: A 3-query non-adaptive PCP with perfect completeness |
| 4 |
2016.09.14 |
Boolean Functions 3 testing linear transformations regime of large agreement / small distance black-box use of BLR with union bound directly via majority decoding [V11, Section 7] regime of small agreement / large distance statement of Samorodnitsky's theorem [V11, Section 1] Balog-Szemerédi-Gowers Theorem [V11, Section 3] (proof skipped in class) Rusza Theorem [V11, Section 4] decoding to a linear transformation [V11, Section 5] |
Main: Viola 2011: Selected results in additive combinatorics: an exposition Additional: Samorodnitsky 2007: Low-degree tests at large distances Lovett 2012: An exposition of Sanders quasi-polynomial Freiman-Ruzsa theorem Lovett 2010: Equivalence of polynomial conjectures in additive combinatorics Lovett 2013: Additive combinatorics and its applications to TCS Trevisan 2007: The unreasonable effectiveness of additive combinatorics in CS |
| 5 |
2016.09.21 |
Real Functions testing monotonicity adaptive tester for the line [EKKRV99, Section 2.1.2] non-adaptive tester for the line [EKKRV99, Section 2.1.1] testing submodularity [SV10] |
Main: Ergün Kannan Kumar Rubinfeld Viswanathan: Spot checkers Seshadhri Vondrák 2010: Is submodularity testable? More on testing monotonicity: Goldreich Goldwasser Lehman Ron Samorodnitsky 1999: Testing monotonicity Khot Minzer Safra 2015: On monotonicity testing and boolean isoperimetric type theorems Testing real functions for Lp distances: Berman Raskhodnikova Yaroslavtsev 2014: Lp-testing Other cool topics about boolean functions that we do not have time to cover: Testing k-linearity Blais Kane 2012: Testing properties of linear functions Majority is stablest Khot Kindler Mossel O’Donnell 2007: Optimal inapproximability results for MAX‐CUT and other 2‐variable CSPs? Mossel O’Donnell Oleszkiewicz 2010: Noise stability of functions with low influences: invariance and optimality (video) Mossel 2013: Majority is Stablest: Discrete and SOS |
| 6 |
2016.09.28 |
Low-Degree Polynomials 1 definition of low-degree testing univariate polynomials d+2 random points (any ε) [S92, Section 3.1.1] d+2 random evenly-spaced points (ε ~ 1/d2) [S92, Section 3.1.2] multivariate polynomials total degree via random lines (ε ~ 1/d2) [S92, Section 3.2.1] bivariate polynomials individual degree via random row/column (ε ~ 1/d) [S92, Section 3.2.2] |
Main: Rubinfeld Sudan 1996: Robust Characterizations of Polynomials with Applications to Program Testing Goldreich 2016: Lecture notes on low-degree tests Additional: Sudan 1992: Efficient checking of polynomials and proofs and the hardness of approximation problems (video) Sudan 2015: Robust low-degree testing Gemmel Lipton Rubinfeld Sudan Wigderson 1991: Self-testing/correcting for polynomials and for approximate functions Friedl Sudan 2010: Some improvements to total degree tests Initial uses of low-degree testing for probabilistic checking: Babai Fortnow Lund 1990: Non-deterministic exponential time has two-prover interactive protocols MIP=NEXP Babai Fortnow Levin Szegedy 1991: Checking computations in polylogarithmic time PCP=NEXP |
| 7 |
2016.10.05 |
Low-Degree Polynomials 2 connection from total degree to individual degree redoing multivariate, but via bivariate (ε ~ 1/d) [S92, Section 3.2.3] bivariate polynomials (tighter analysis via Bézout) individual degree via random row/column (ε ~ constant) [PS94, Sections 3,4,5] |
Main: Rubinfeld Sudan 1996: Robust Characterizations of Polynomials with Applications to Program Testing Polishchuk Spielman 1994: Nearly linear-size holographic proofs Additional: Spielman 1995: Computationally efficient error-correcting codes and holographic proofs Matoušek 2012: The weak Bézout theorem Guth 2012: Bézout theorem |
| 8 |
2016.10.12 |
Low-Degree Polynomials 3 low-degree testing at large distances [RS97] random planes tester [H10, Lecture 7, Section 7.1] equivalence of random planes and plane-vs-point [H10, Lecture 7, Section 7.2] analysis of plane-vs-point [H10, Lecture 8] |
Main: Lecture 7 of Harsha's 2010 course Lecture 8 of Harsha's 2010 course Raz Safra 1997: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP Additional: Raz Moshkovitz 2008: Sub-constant error low degree test of almost linear size |
| 9 |
2016.10.19 |
Tensor Product Codes testing tensor products of codes definition of tensor product codes [V11, Section 2.1] definition of locally testable codes [V11, Section 2.2] random axis-parallel hyperplane tester [V11, Appendix A] random axis-parallel plane tester [V11, Appendix C.1] |
Main: Ben-Sasson Sudan 2006: Robust locally testable codes and products of codes Viderman 2011: A combination of testability and decodability by tensor products Positive results for robust testability 2-wise tensor products: Dinur Sudan Wigderson 2006: Robust local testability of tensor products of LDPC codes Ben-Sasson Viderman 2009: Composition of semi-LTCs by two-wise tensor products Ben-Sasson Viderman 2009: Tensor products of weakly smooth codes are robust Meir 2011: On the rectangle method in proofs of robustness of tensor products Negative results for robust testability 2-wise tensor products: Valiant 2005: The tensor product of two codes is not necessarily robustly testable Goldreich Meir 2007: The tensor product of two good codes is not necessarily robustly testable Coppersmith Rudra 2005: On the robust testability of product of codes Other cool topics that we do not have time to cover: Testing direct products Dinur Seurer 2013: Direct product testing Impagliazzo Kabanets Wigderson 2009: New direct-product testers and 2-query PCPs Impagliazzo Jaiswal Kabanets Wigderson 2010: Uniform direct product theorems: simplified, optimized, and derandomized Polynomially low error Moshkovitz 2016: Low degree test with polynomially small error Bhattacharyya Kopparty Schoenebeck Sudan Zuckerman 2010: Optimal testing of Reed-Muller codes Alon Kaufman Krivelevich Litsyn Ron 2005: Testing Reed-Muller codes |
| 10 |
2016.10.26 |
Lower Bounds via communication complexity for k-linearity [G16, Section 3] by showing YES and NO distributions paradigm [G16, Section 2] univariate polynomials (1- and 2-sided) 1-sided graph regularity monotonicity over hypercubes [FLNRRS02, Section 7] |
Main: Goldreich 2016: Notes for lower bounds techniques Blais Brody Matulef 2012: Property testing lower bounds via communication complexity Fischer Lehman Newman Raskhodnikova Rubinfeld Samorodnitsky: Monotonicity testing over general poset domains Additional: Bogdanov Trevisan 2004: Lower bounds for testing bipartiteness in dense graphs |
| X |
2016.11.02 |
No class. |
No class. |
| 11 |
2016.11.09 |
Graph Properties 1 introduction of the dense graph model definitions [GGR98, Section 5] testing bipartiteness sampling O(1/ε2) vertices [GGR98, Section 6.1] testing k-colorability sampling O(k*log(k)/ε2) vertices [AK02, Section 4] |
Main: Goldreich Goldwasser Ron 1998: Property testing and its connection to learning and approximation Alon Krivelevich 2002: Testing k-colorability More on upper and lower bounds for bipartiteness: Bogdanov Trevisan 2004: Lower bounds for testing bipartiteness in dense graphs Bogdanov Li 2010: A better tester for bipartiteness? |
| 12 |
2016.11.16 |
Graph Properties 2 Szemerédi’s regularity lemma statement [V, Section 1] proof [V, Section 3] (watch out for typos) [L, Section 1] testing triangle freeness triangle removal lemma [V, Section 2.3] [L, Section 3] distinguishing few vs many triangles [GS14, Section 3.4] |
Main: Szemerédi 1975: Regular partitions of graphs Verstraëte: Regularity lemma (lecture notes) Lee: Regularity lemma (lecture notes) Additional: Alon Fischer Krivelevich Szegedy 2000: Efficient testing of large graphs Alon 2002: Testing subgraphs in large graphs Alon Fischer Newman Shapira 2009: A combinatorial characterization of the testable graph properties: it's all about regularity Fox 2010: _ A new proof of the graph removal lemma_ Alon Fox 2015: Easily testable graph properties |
| X |
2016.11.23 |
No class. |
No class. |
| 13 |
2016.11.30 |
Research project presentations |
Schedule: 1310-1330: HKN Evaluations 1330-1350: Peter Manohar, m-variate low degree testing 1350-1410: Yuting Wei, testing shape restriction of discrete distribution 1410-1430: Aaron Schild, local computation algorithms 10-min break 1440-1500: Pratyush Mishra, PCP proof composition 1500-1520: Pasin Manurangsi, (2-eps) vertex cover hardness 1520-1540: Xingyou Song, submodular testing and optimization 1540-1600: Daniel Shaw, a combinatorial characterization of testable graph properties and regularity later on Benjamin Weitz, another proof of the BLR test: generalizing to different domains |
| X |
2016.12.07 |
No class. |
No class. |