In this course we study several of recent papers in the field of approximation algorithms.
We focus on many of the underlying techniques including randomized rounding,
spectral methods, random walks and real stable polynomials.
Administrative Information
- Instructor: Shayan Oveis Gharan
- Office Hours: By appointment, email me at shayan at cs dot washington dot edu.
- Lectures: Monday - Wednesday 4:00 - 5:20 at EEB 031
Assignments
- Problem Set 1, due date April 8th in class.
- Problem Set 2, due date April 20th in class.
- Problem Set 3, due date May 11th in class.
- Problem Set 4, due date June 8th in class.
Related Materials
- Luca Trevisan's lecture notes on Spectral Graph Theory
- Dan Spielman's lecture notes on Spectral Graph Theory
- Jan Vondrak's lecture notes on Combinatorial Optimization
- Eigenvalues of Graphs by Laszlo Lovasz.
- Algebraic Graph Theory by Chris Godsil.
- Pablo Parrilo's lecture notes on Algebraic Techniques and Semidefinite Optimization
| Lecture | Topic | Notes | Bonus Materials |
|---|---|---|---|
| Lecture 1 (03/30/15) | Background, Matrix Tree Theorem | tex, pdf | |
| Lecture 2 (04/01/15) | Negative Correlation and Burton and Pemantle Theorem | ||
| Lecture 3 (04/06/15) | Characteristic Polynomials and proof of Burton,Pemantle Theorem | tex, pdf | |
| Lecture 4 (04/08/15) | Electrical Flows and Effective Resistance | tex, pdf | |
| Lecture 5 (04/13/15) | Effective Resistance and Random Walks | ||
| Lecture 6 (04/15/15) | An O(logn/loglogn) Approximation Algorithm for Asymmetric TSP | tex, pdf | |
| Lecture 7 (04/20/15) | |||
| Lecture 8 (04/22/15) | |||
| Lecture 9 (04/27/15) | The Pipage Rounding Method | tex, pdf | |
| Lecture 10 (04/29/15) | Hyperbolic and Real Stable Polynomials | tex, pdf | |
| Lecture 11 (05/04/15) | Interlacing and Strongly Rayleigh distributions | ||
| Lecture 12 (05/06/15) | A Randomized Rounding Algorithm for STSP | tex, pdf | |
| Lecture 13 (05/11/15) | A Construction of Linear size Thin Forests | tex, pdf | |
| Lecture 14 (05/13/15) | Spectrally Thin Subsets using Interlacing Polynomials | tex, pdf | |
| Lecture 15 (05/18/15) | Kadison-Singer Conjecture and its Extensions | tex, pdf | |
| Lecture 16 (05/20/15) | Proof of the Kadison Singer Conjecture | ||
| Lecture 17 (06/01/15) | Cheeger's Inequality and the Sparsest Cut Problem | tex, pdf | |
| Lecture 18 (06/03/15) | Existence of Spectrally Thin Trees | tex, pdf | |
| Lecture 19 (05/08/15) | The Dual of Effective Resistance CP |
Tentative Agenda
- Random Spanning Trees and Determinantal Measures
- Electrical Flows and Effective Resistance
- Rounding by Sampling Method and O(log n/loglog n) approximation algorithm for ATSP
- Randomzied Rounding by Pipage Rounding Method
- Randomized Rounding Algorithm for the Steiner Tree Problem
- Real Stable and Hyperbolic Polynomials
- Strongly Rayleigh Distributions and their properties
- Approximation Algorithms for Symmetric TSP
- Interlacing Polynomials and the Bourgain-Tzafriri Theorem
- Kadison Singer Problem and the Thin Basis Problem
- Thin Edge Covers and Improved Approximation Algorithms for Asymmetric TSP
- Spectral Graph Theory and Cheeger Inequality
- Subspace Enumeration Method and Subexponential time algorithms for Unique Games Problem
- Higher Order Cheeger Inequality
- Sum of Squares and Approximation Algorithms for Low Threshold Rank Graphs
