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I have always seen the notion of nondeterminism in computation attributed to Michael Rabin and Dana Scott. They defined nondeterministic finite automata in their famous paper Finite Automata and Their …

Applications of Ramsey theory to CS, Gasarch

There is a combinatorial algorithm by Mahajan and Vinay that works over commutative rings: http://cjtcs.cs.uchicago.edu/articles/1997/5/contents.html

Check Chapter 7 of Salil Vadhan's monograph. Corollary 7.64 is Impagliazzo and Wigderson's result.

There are many links between discrepancy theory and computer science, and Bernard Chazelle has beautifully surveyed some of them in his book. A number of links have been found more recently as well, f …

Universal algebra is an important tool in studying the complexity of constraint satisfaction problems. For example, the Dichotomy Conjecture states that, roughly speaking, a constraint satisfaction …

(Disclaimer: I am not an expert, feel free to suggest corrections, or write a more comprehensive answer if you are.) Extending computability and complexity to the real numbers (which is a first step …

Notice that approximating sparsest cut to within $\alpha$ gives a $2\alpha$ approximation for the Cheeger constant as defined. Here are some papers that give constant approximation algorithms for spar …

NP is equal to coNP if and only if there are efficiently verifiable proofs of unsatisfiability. I.e., if and only if there exists a polynomial time turing machine $M$, which given any SAT formula $\ph …

The Batson-Spielman-Srivastava barrier function method has had a number of applications to geometry and functional analysis, arose in computer science, and is a very original form of potential functio …

I didn't mean this to be an answer but it would require too many comments. Hope it's useful. As Tsuyoshi points out, it's tempting to conjecture that all "non-trivial" properties are hard (NP-hard fo …

A relatively simple setting to illustrate the method of random restrictions is Subbotovskaya's original application of the method to prove an $\Omega(n^{1.5})$ lower bound on the formula size of the p …

There are variants of Berry-Esseen for bounded independence, although I have not seen one which is as general as the original theorem. For example Theorem 5.1. in Diakonikolas, Kane, Nelson implies a …

Kearns and Vazirani is maybe a bit old, but good introduction.

I do not think this is in AC0 and I can show a lower bound for the related promise problem of distinguishing between $\sum x_i = 0$ and $\sum x_i = 2$, when $x \in \{-1, 1\}^n$. Similar Fourier techni …