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This is a branch that includes: computational complexity theory; complexity classes, NP-completeness and other completeness concepts; oracle analogues of complexity classes; complexity-theoretic computational models; regular languages; context-free languages; Komolgorov Complexity and so on.

26 votes
6 answers
7k views

Are there any interesting examples of random NP-complete problems?

Here's an example of the kind of thing I mean. Let's consider a random instance of 3-SAT, where you choose enough clauses for the formula to be almost certainly unsatisfiable, but not too many more th …
gowers's user avatar
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15 votes
3 answers
1k views

Is this strange problem NP-complete?

The following quadratic expression can be simplified: (x+1)(x+2) + (x+1)(x-3) + 2x(2x-1) - (3x+1)(x-3) - 2x(x+2). What is the easiest way of doing the simplification? (It would be good to think abou …
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29 votes
2 answers
1k views

A combination of two well-known complexity problems

Suppose you are given two graphs $G$ and $H$ and are told that one of the following two situations occurs. Either they are isomorphic, or one of the graphs contains a Hamilton cycle and the other does …
gowers's user avatar
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23 votes
2 answers
2k views

What is the complexity of this problem?

Recently on Dick Lipton and Ken Regan's blog there was a post about problems of intermediate complexity, that is, NP problems that are harder than P but easier than NP-complete. The main message of th …
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12 votes
1 answer
575 views

Are there very strongly pseudorandom permutations?

A pseudorandom permutation can be defined formally as a function $\phi$ from $\{0,1\}^k\times\{0,1\}^n$ to $\{0,1\}^n$ such that for every $x\in\{0,1\}^k$ the function $\phi_x:y\mapsto\phi(x,y)$ is a …
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