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Course10610622963 lectures

Computational Complexity

Prof. Subrahmanyam Kalyanasundaram · IIT Hyderabad

Start with lecture 1 At NPTEL

About

This course is an introduction to the area of computational complexity theory. We will see different models of computations and computational complexity classes. The computational models measure various different aspects of computation, like time, space, randomness, number of gates, amount of communication etc. The complexity classes classify different computational problems depending on their easiness or hardness as per these different models. We will also see how these complexity classes are related to each other. Many of the results are extremely interesting and use several interesting ideas.

Intended audience: Students of Computer Science discipline. This could be BTech students who are interested in Theory or MTech/PhD students.

Prerequisites: Theory of Computation.

Lectures

Week 1

  1. Introduction to Computational Complexity YouTube
  2. The Class P YouTube
  3. The Class NP YouTube
  4. The Class NP - Alternate Definition YouTube
  5. Polynomial Time Reductions YouTube
  6. NP - Completeness YouTube

Week 2

  1. Cook Levin Theorem: Part 1 YouTube
  2. Cook Levin Theorem: Part 2 YouTube
  3. More NP Complete Problems YouTube
  4. Polynomial Hierarchy: Part 1 YouTube
  5. Polynomial Hierarchy: Part 2 YouTube
  6. Polynomial Hierarchy: Part 3 YouTube

Week 3

  1. Time Hierarchy Theorem YouTube
  2. Introduction to Space Complexity YouTube
  3. NL-Completeness YouTube
  4. Savitch's Theorem YouTube
  5. NL = co-NL: Part 1 YouTube
  6. NL = co-NL: Part 2 YouTube

Week 4

  1. PSPACE Completeness YouTube
  2. Games and PSPACE Completeness YouTube
  3. Space Hierarchy Theorem YouTube
  4. Ladner's Theorem YouTube
  5. Oracle Turing Machines YouTube
  6. Polynomial Hierarchy Using Oracles YouTube

Week 5

  1. Baker-Gill-Solovay Theorem: Part 1 YouTube
  2. Baker-Gill-Solovay Theorem: Part 2 YouTube
  3. Randomized Complexity Classes: Part 1 YouTube
  4. Randomized Complexity Classes: Part 2 YouTube
  5. Randomized Complexity Classes: Part 3 YouTube

Week 6

  1. Randomized Complexity Classes: Part 4 YouTube
  2. Comparison Between Randomized Complexity Classes YouTube
  3. BPP is in Polynomial Hierarchy YouTube
  4. Circuit Complexity: Part 1 YouTube
  5. Circuit Complexity: Part 2 YouTube

Week 7

  1. Formal Definition of Circuits YouTube
  2. Hierarchy Theorem for Circuit Size YouTube
  3. Complexity Class : P/Poly YouTube
  4. Karp-Lipton Theorem YouTube
  5. Turing Machines That Take Advice YouTube
  6. Classes NC and AC YouTube

Week 8

  1. Parity Not in AC0: Part 1 YouTube
  2. Parity Not in AC0: Part 2 YouTube
  3. Adleman's Theorem YouTube
  4. Polynomial Identity Testing and Bipartite Perfect Matching in RNC YouTube

Week 9

  1. Search Bipartite Perfect Matching is in RNC: Part 1 YouTube
  2. Search Bipartite Perfect Matching is in RNC: Part 2 YouTube
  3. Promise Problems and Valiant-Vazirani Theorem YouTube
  4. Valiant Vazirani Theorem Continued YouTube
  5. #P and the Complexity of Counting YouTube

Week 10

  1. Permanent is #P-Complete: Part 1 YouTube
  2. Permanent is #P-Complete: Part 2 YouTube
  3. Toda's Theorem: Part 1 YouTube
  4. Toda's Theorem: Part 2 YouTube

Week 11

  1. Introduction to Communication Complexity: Part 1 YouTube
  2. Introduction to Communication Complexity: Part 2 YouTube
  3. Lower Bound Techniques YouTube
  4. Communication Complexity of Relations YouTube
  5. Monotone Depth Lower Bound for Matching YouTube

Week 12

  1. Interactive Proofs YouTube
  2. #3SAT is in IP YouTube
  3. Public Coin Interactive Proofs and AM/MA YouTube
  4. Simulating Private Coins using Public Coins YouTube
  5. Summary and Concluding Remarks YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Hyderabad
Department
Computer Science and Engineering
Teacher
Prof. Subrahmanyam Kalyanasundaram
Level
Graduate
Language
English
Subjects
Computer science, Engineering
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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