### Computational Complexity Theory - Monsoon 2018

Credits: 4

Pre-requisites: Algorithms, Discrete Maths, Theory of Computing, Probability

Post-conditions: Upon successful completion of the course, the student will gain the following:
• knowledge of different models of computation and their relationships

• ability to bound resource usage (e.g., time, space, circuit size, depth) required for problems under different models of computation

• ability to place a problem in the hierarchy of computational complexity classes

• ability to show separations, collapses of classes and prove completeness of problems

• understand the major questions (e.g., P =? NP) and their role in other fields of computer science

• (if time & scope permits) develop knowledge of randomized complexity classes and ability to perform randomized reductions

## Description

Computational problems can be studied from the point of view of computational resources (e.g., running time) required to solve them – this forms the notional of computational difficulty. Apart from the nature of the problem, the difficulty also depends on the underlying model of computation. Problems can be related to each other based on their resource usage, and problems with similar difficulty can be grouped together to form a classification of computational problems into complexity classes.

Computational complexity is about studying the above concepts, and is especially concerned with giving precise upper and lower bound on the amount of resources required to solve certain problems. This has had a profound impact on current algorithm design and cryptography, and still sees applications in areas outside of theoretical computer science.

### Instruction

Lectures will follow a mixture of regular lectures and flipped-classroom model. Hence, maturity in logic and reasoning is mandatory. The course will start where a standard course on Theory of Computation ends, hence it is strongly advisable that students are comfortable and strong in TOC.

Book
• [AB] Sanjeev Arora and Boaz Barak. “Complexity Theory: A Modern Approach”
• (Reference) [S] Michael Sipser, “Introduction to the Theory of Computation”
Evaluation: 75% exams, 25% course-work and home-work
• 40% final exam
• 30% mid-sem
• 5% quiz / scribe (see below)
• 18% homework (1% for each homework problem, lowest 15% problems to be dropped)
• 5% paper reading (to be decided after mid-sem exam)
• 2% - free (thanks for attending the course)

Tentative schedule

 Lecture Topics Chapter Lec 1,2 Det. Turing machine definitionLinear speedupPalindrome lower bound AB1  Lec0 Lec1Lec2 by Parth Lec 3 NDTM tape reduction, linear speedup Lec3 by Siddhartha Lec 4 Space complexity, space compression, time and space simulations.Hierarchy of common complexity classes. Lec4 by Lavina Lec 5 Non-deterministic space classes, Savitch's theorem Lec5 by Hanit Lec 6 Sub-logarithmic space TMs Lec6 Lec 7 Reduction, Complete problems Lec7 by Niket Lec 8 Paper discussion: DTIME^1(o(n log n)) = REG, Tally languages, Padding techniques, Log-space reduction, complete problems Theorem 3.3 in Koboyashi1985Lec8 (pending) Lec 9 Space hierarchy theorem Lec9 by Sushant Lec10 Time hierarchy theorem, NDTM hierarchy theorem AB3 Lec10 by Harshit Lec11 Turing reduction, BGS theorem AB3 Lec11 by Siddharth Lec12 Ladner's theorem Lec12 AB3 Lec13 PH AB5 Lec14 Paper presentation by Siddhartha & Hanit Paper Lec15 PH-completeness and coNL-completeness AB4, AB5 Lec16 NL and coNL. Boolean circuit complexity. AB4, AB6 Lec17 P/poly AB6 Lec18 Circuit size upper bounds and Shannon lower bound, Karp-Lipton AB6 Lec19 Paper presentation by Varun Paper Lec20 Size-hierarchy theorem, Kannan's theorem AB6 notes by Paul Beame Lec21 Paper presentation by Parth Paper Lec22 NC and AC circuit classes AB6 Lec23 NC and Logspace classes, Furst-Saxe-Sipser AB6 Lec24 Probabilistic complexity classes AB7.1,7.3 Lec25 Paper presentation by Niket Paper Lec26 Paper presentation by Sushant Paper

Scribe instructions

Rename ex.tex to LecXX.tex. Do not modify preamble.tex.
Try to present the results as reusable lemma/theorem/etc. and use figures if that improve clarity (use ipe/xfig/etc. pdf-producing WYSIWIG tools).
Subpages (1):
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ex.tex
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Debajyoti Bera,
Aug 18, 2018, 11:08 AM
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preamble.tex
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Debajyoti Bera,
Sep 4, 2018, 2:51 AM