Chapter 7: Property testing, PCPPs, and CSPs « Analysis of Boolean Functions

Chapter 7 notes

The study of property testing was initiated by Rubinfeld and Sudan [RS96] and significantly expanded by Goldreich, Goldwasser, and Ron [GGR98]; the stricter notion of local testability was introduced (in the context of error-correcting codes) by Friedl and Sudan [FS95]. The first local tester for dictatorship was given by Bellare, Goldreich, and Sudan [BGS95,BGS98] (as [...]

Chapter 7 exercises

[...]

§7.4: Highlight: Håstad’s hardness theorems

In Theorem 36 we saw that it is $\mathsf{NP}$-hard to $(1-\delta_0, 1)$-approximate Max-E$3$Sat for some positive but inexplicit constant $\delta_0$. You might wonder how large $\delta_0$ can be. The natural limit here is $\frac18$ because there is a very simple algorithm which satisfies a $\frac78$-fraction of the constraints in any Max-E$3$Sat instance:

[...]

§7.3: CSPs and computational complexity

This section is about the computational complexity of constraint satisfaction problems (CSPs), a fertile area of application for analysis of boolean functions. To study it we need to introduce a fair bit of background material; in fact, this section will mainly consist of definitions.

[...]

§7.2: Probabilistically checkable proofs of proximity

In the previous section we saw that every subproperty of the dictatorship property has a $3$-query local tester. In this section we will show that any property whatsoever has a $3$-query local tester — if an appropriate “proof” is provided.

[...]

§7.1: Dictator testing

In Chapter 1.6 we described the BLR property testing algorithm: given query access to an unknown function $f : \{0,1\}^n \to \{0,1\}$, this algorithm queries $f$ on a few random inputs and approximately determines whether $f$ has the property of being linear over ${\mathbb F}_2$. The field of property testing for boolean functions is concerned [...]