Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Prerequisites
MIT students need permission of instructor.
Course Description
The topic of this course is Probabilistically Checkable Proofs (PCPs), with emphasis on their applications in hardness of approximation.
It is firmly believed that the class of NP-hard problems does not admit polynomial time algorithms. For some NP-hard problems, the task of finding an approximation algorithm, that is, an algorithm that finds a solution which is “close” to the optimum, remains NP-hard. Such results are known as hardness of approximation results, and proving such statements requires a fundamental Theoretical Computer Science (TCS) result known as the PCP theorem. This theorem has its origins in proof systems, has close connections to optimization, combinatorics, Fourier analysis and more, and has wide array of applications throughout TCS.
In this course, we will present the theory of PCPs, and prove some fundamental consequences of it as well as more recent advances. More specifically, the first half of the course will be devoted to the (algebraic) proof of the basic PCP Theorem and basic relation to approximation problems. We will then move on to more advanced topics, such as hardness amplification, the long-code framework, the Unique-Games Conjecture and its implications, and the 2-to-2 Games Theorem.
Material from other courses in Probabilistically Checkable Proofs can be found online in various places; for example:
- Hardness of Approximation: PCP Theorem to 2-to-2 Games Theorem online course by Subhash Khot, NYU
- Probabilistically Checkable Proofs online course by Irit Dinur and Dana Moshkovitz, Weizmann Institute of Science
- Foundations of Probabilistic Proofs online course by Alessandro Chiesa, Berkeley
Grading
There will be 5 problem sets during the course, and the final grade in the course will be an average of your grade in the problem sets.
