## CSCI 6114: Computational Complexity (Grad, Fall 2021)

### Syllabus

### Online Resources

- Complexity Zoo
- TCS StackExchange
- ECCC (like arXiv's cs.CC and cs.DS categories, but higher-quality; people often cross-post here and arXiv)
- arXiv cs.CC and cs.DS
- DBLP
- Prof. Grochow's Mind Map of (part of) Complexity - for helping explore & pick class topics

### Books

There is no textbook for this class, we will read selections from some subset of the books below, lecture notes, and research articles. However, regardless of what we cover in class, these books are great complexity resources! (When available through CU libraries the link below is to the CU Library Proxy that will log you in; if you don't want that look at the URL - it should be fairly clear how to modify it to not do that, or ask me for help)

Some of my favorite non-textbooks

- Wigderson. Mathematics & Computation: A theory revolutionizing technology and science (pdf draft available at link)
- Hemaspaandra & Ogihara. The Complexity Theory Companion. Available digitally through CU libraries here
- Selman (editor). Complexity Theory Retrospective.
- Schöning and Pruim. Gems of Theoretical Computer Science

Some good textbooks

- Arora & Barak. Computational Complexity: A Modern Approach (the draft pdf from their website is fine)
- Du & Ko. Theory of Computational Complexity.
- Köbler, Schöning, and Torán, The Graph Isomorphism Problem: Its Structural Complexity. Even though it is focused on GI, it covers a fair bit of just general complexity, including oracles, PH, circuits, randomized classes, interactive proofs, and more.
- For more math behind Graph Isomorphism, check out Lauri & Scapellato, Topics in Graph Automorphisms and Reconstruction
- Some less advanced but really excellent textbooks whose later chapters might make appearances in this class: Sipser Ch. 8-10; Moore & Mertens Ch. 8,10,11; Homer & Selman Ch. 5,7-12

### In-Class Exercises

Classes 2-3: Circuits and P/polyClasses 4-6: Polynomial Hierarchy PH

Class 6-8: Oracles, relativization, and the polynomial hierarchy

Class 9-11: Graph Isomorphism

Class 11-13: Weisfeiler—Leman

Class 14-16: Proof complexity

Classes 16-19: Algebraic proof complexity

Classes 20-23: Randomized complexity

Classes 24-26: PSPACE and IP

### Final Projects

These projects were chosen by the students in the class, to be presented during the last two weeks of term:- Ladner's Theorem
- Reductions between tautologies
- Barrington's Theorem
- Quantum complexity
- The Raz-Tal oracle separating BQP from PH
- Stochastic parity games
- The power of MOD6 circuits
- Why Do Local Methods Solve Nonconvex Problems?