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# Computational Complexity

IIT Hyderabad and NPTEL via Swayam

## Overview

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.

## Syllabus

Week-1: Introduction to the course, Review of NP Completeness, P vs NP, Cook-Levin Theorem
Week-2: Time Hierarchy Theorem, Polynomial Hierarchy, Introduction to Space Complexity
Week-3: Savitch’s Theorem, NL-Completeness, NL = coNL Week-4: PSPACE Completeness, Space Hierarchy Theorem, Baker-Gill-Solovay Theorem
Week-5: Randomized Complexity Classes, BPP is in polynomial hierarchy
Week-6:Nonuniform computation, Circuit Complexity
Week-7: Parity not in AC^0
Week-8: Karp-Lipton Theorem, Adleman’s Theorem, Polynomial Identity Testing, Isolation Lemma, Perfect Matching is in NC^2
Week-9: #P and #P Completeness. Permanent is #P Complete.
Week-10: Valiant Vazirani Theorem and Toda’s Theorem
Week-11: Communication Complexity, Monotone depth lower bound for matching
Week-12: Interactive Proofs.

### Taught by

Prof. Subrahmanyam Kalyanasundaram

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