CS366: Graph Partitioning and Expanders
[general info]
[lecture notes] [exams and projects]
what's new
general information
Instructor: Luca
Trevisan, Gates 474, Tel. 650 723-8879, email trevisan at stanford dot edu
Classes are Monday-Wednesday, 11am-12:15pm, in 120-59
Office hours: Wednesdays 2-3pm
About the course
The mathematics of expander graphs is studied by three distinct
communities:
- The algorithmic problem of finding a small balanced cut
in a graph (that is, of finding a certificate that a graph is *not* an
expander) is a fundamental problem in the area of approximation
algorithms, and good algorithms for it have many applications, from
doing image segmentation to driving divide-and-conquer procedures.
-
Explicit constructions of highly expanding graphs have many applications
in algorithms, data structures, derandomization and cryptography; many
constructions are algebraic, and lead to deep questions in group theory,
but certain new constructions are purely combinatorial.
- The speed of
convergence of MCMC (Markov-Chain Monte-Carlo) algorithms is related to
the expansion of certain exponentially big graphs, and so the analysis
of such algorithms hinges on the ability to bound the expansion of such
graphs.
In this course we aim to present key results from these three areas, and
to explore the common mathematical background.
Prerequisites: undergraduate-level understanding of discrete probability, linear algebra, and
algorithms; preferably, also a basic understanding of linear programming
and of duality.
Assignments: a midterm take-home exams and a take-home final exam.
Working on a research project related to the topics of the class can
substitute for the final exam.
References
The main reference will be a set of lecture notes. Notes will be
posted after each lecture. In addition, the following texts
will be helpful references.
On sparsest cut approximation algorithms:
On spectral graph theory and on explicit constructions
of expander graphs:
On Markov-Chain Monte-Carlo algorithms for uniform generation
and approximate counting.
The following is a tentative schedule:
- Definitions: edge and vertex expansion, uniform and general sparsest
cut problems, review of linear algebra
- Eigenvalues and expansion, Cheeger's inequality and the spectral
partitioning algorithm
- Cheeger's inequality, continued
- Classes of graphs for which spectral partitioning is provably good
- Algorithms for finding sparse cuts: Leighton-Rao, and metric embeddings
- Equivalence of rounding the Leighton-Rao relaxation and
embedding general metrics into L1
- Algorithms for finding sparse cuts: Arora-Rao-Vazirani
- Arora-Rao-Vazirani, continued
- Integrality gaps for the Arora-Rao-Vazirani relaxation
- Applications of expanders: derandomization
- Applications of expanders: security amplification of one-way permutations
- The Margulis-Gabor-Galil construction of expanders
- The Zig-Zag graph product construction
- Eigenvalues, expansion, conductance, and random walks
- Approximate counting, approximate sampling, and the MCMC method
- Random Spanning trees
- Counting colorings in bounded-degree graphs
- Counting perfect matchings in dense bipartite graphs
- The Metropolis algorithm
classes and lecture notes
- Jan 7. Introduction. Basics of linear algebra and introduction to spectral graph theory
- Jan 9. Cheeger inequality
- Jan 14. Cheeger inequality, continued
- Jan 16. The eigenvalues and eigenvectors of Cayley graphs
Jan 21. No class - MLK day
- Jan 23. The Leighton-Rao relaxation
- Jan 28. Bourgain's theorem
- Jan 30. The power method to find approximate eigenvectors
Feb 4. No class - Luca is away
Feb 6. No class - Luca is away
- Feb 11. Semidefinite programming and the Arora-Rao-Vazirani relaxation
- Feb 13. Rounding the Arora-Rao-Vazirani relaxation
Feb 18. No class - President's day
- Feb 20. The problem of finding well-separated sets
- Feb 25. Finding well-separated sets
- Feb 27. Other results and open problems in spectral graph theory
- Mar 4. Gabber-Galil expanders
[notes]
midterm and final