Computational Complexity (Spring 2017)

This course describes attempts to answer a fundamental question of CS namely: What can computers solve efficiently?

Some Bibliography:

While the course does not follow one book in a precise order, most of the material can be found in the following two books (that are available online). I will point to relevant section in the books when teaching the material.

Some of the topics (at least in the beginning of the course) are also covered in the following books.

Requirements:

There will be an exam at the end of the course. During the lectures I will sometimes present questions and challenges for you to think about. You are encouraged to work on these questions.

Exercises:

These exercises are not for submission. However you are strongly encouraged to solve them.

-        Exercise 1 (on space complexity)

-        Exercise 2 (on diagonalization and the polynomial time hierarchy)

-        Exercise 3 (on circuits and randomized algorithms)

-        Examples of class exams: (1) (2) (3) (4)

 

Take Home Exam:

If you came to class regularly, you may ask for permission to do a take home exam (instead of an exam, you cannot do both).

Here is the take home exam.

If you are interested in taking the take home exam, please write to me and send me:

1.    Your name and ID.

2.    How many lectures did you miss?

3.    Why you want to do a take home exam, and not a class exam.

You must receive an approval from me in order to do the take home exam.

Material:

 

o   The generality of the argument: the deterministic space hierarchy theorem.

o   Why the argument doesn't give a nondeterministic time hierarchy theorem.

o   The non-deterministic time hierarchy theorem (delayed diagonalization).

o   Weakness of diagnolization: results proven this way hold for any oracle.

o   An oracle A such that NP^A <> P^A. Implies impossibility of proving NP<>P by direct diagonalization.

o   The notion of relativizing statements and proofs.

o   The statement NP=P and NP<>P are non-relativizing.

 

       The polynomial time hierarchy

o   Definition with quantifiers and definition with oracles (equivalence is an exercise).

o   The hierarchy collapses if PH has a complete problem.

o   The hierarchy collapses if Sigma_i = Pi_i.

o   Padding arguments.

o   Time space tradeoffs: NTIME(n) not contained in TISP(n^{1.2},n^{0.2}).

 

       Circuits

 

 

        Probabistically checkable proofs

o   Definition of the class PCP(r,q).

o   Proof that NP is in PCP(poly(n),O(1)).