Computer Science
Dartmouth College

### Fall 2011

Instructor: Amit Chakrabarti, Sudikoff 107, Office hours: MF 9:00-10:30

Location:  Sudikoff 214 (or 245)

When:  2A hour: TTh 14:00-16:00 X-hr W 16:15-17:20

What:  We shall study complexity theory through concrete computational models, of which the primary examples are decision trees, circuits, and communication protocols. In other words, we will study query complexity, circuit complexity, and a bit of communication complexity. For this term's offering, we shall not focus on structural complexity.

Prerequisites:  The main prerequisites are

1. previous exposure to theoretical CS, for example through Dartmouth's CS 39, and
2. a good mathematical background, because we'll study and prove many theorems.
If you are unsure about whether you meet the prerequisites, please see the professor.

#### Outline of Topics

• The complexity of sorting and selection
• Boolean decision trees and basic analysis of Boolean functions
• The complexity of graph properties
• Can algebraic operations help? No, but it's challenging to prove
• Basic circuit lower bounds
• Stronger circuit lower bounds: constant depth
• Still stronger ones: monotone circuits
• Communication protocols, and a connection with circuits
• Algebraic circuits, if time permits

#### Possible Day-by-Day Plan

The plan below is nowhere near final. It is just an example of how the course might unfold. I will likely permute some of the topics and de-emphasize others to make room for new material.

 I. Decision Trees: Lower Bounds Based on Data Access 1. Sep 27 Comparison trees. Complexity of sorting and selection. 2. Sep 29 Algebraic decision/computation trees. Milnor-Thom theorem, Ben-Or's theorem 3. Oct 04 Boolean decision trees, certificate complexity, sensitivity, block sensitivity. 4. Oct 06 Relations between the decision tree complexity and the above measures. 5. Oct 11 Polynomial degree and its relation to the above measures. 6. Oct 13 Symmetric and monotone functions. Symmetry under transitive group. Graph properties. 7. Oct 18 Rivest-Vuillemin theorem. Karp's evasiveness conjecture. 8. Oct 20 Randomized complexity. O'Donnell-Saks-Schramm-Servedio theorem. Graph properties redux. 9. Oct 25 Friedgut-Kahn-Wigderson theorem: complexity of graph properties vs threshold probability. II. Circuits: Lower Bounds Based on Data Movement 10. Oct 27 Circuits, P/poly, Shannon's theorem, Lower bounds for THR2 11. Nov 01 AC0, Hastad's switching lemma 12. Nov 03 Circuits with MOD gates, ACC0, Razborov-Smolensky theorem 13. Nov 08 Monotone circuits, Razborov's great exponential lower bound 14. Nov 10 Slack 15. Nov 15 Branching programs, Barrington's theorem 16. Nov 22 (just before thanksgiving) Communication complexity, some basic results 17. Nov 29 Monotone circuit depth, Karchmer-Wigderson theorem 18. When? Algebraic circuits (time permitting)

#### Homework Sets

There will be 4 homework sets given out at regular intervals during the term. In lieu of a final exam, students taking the course for credit may be asked to do a short (15-minute) presentation on an advanced topic of their choosing. Please see the professor for possible suggestions and reading lists.

If you are taking the course for credit, you should aim for a score of at least 140 points. There will be around 200 points worth of homework in total, throughout the term.

Homework 1, due Oct 21.

Homework 2, due Nov 4.

Homework 3, due Nov 28.

Homework 4, due Dec 6.