CS366: Graph Partitioning and Expanders
[general info]
[lecture notes] [exams and projects]
what's new
general information
Instructor: Luca
Trevisan, Gates 474, Tel. 650 7238879, email trevisan at stanford dot edu
Classes are MondayWednesday, 11am12:15pm, in 12059
Office hours: Wednesdays 23pm
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 divideandconquer 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 (MarkovChain MonteCarlo) 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: undergraduatelevel understanding of discrete probability, linear algebra, and
algorithms; preferably, also a basic understanding of linear programming
and of duality.
Assignments: a midterm takehome exams and a takehome 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 MarkovChain MonteCarlo 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: LeightonRao, and metric embeddings
 Equivalence of rounding the LeightonRao relaxation and
embedding general metrics into L1
 Algorithms for finding sparse cuts: AroraRaoVazirani
 AroraRaoVazirani, continued
 Integrality gaps for the AroraRaoVazirani relaxation
 Applications of expanders: derandomization
 Applications of expanders: security amplification of oneway permutations
 The MargulisGaborGalil construction of expanders
 The ZigZag graph product construction
 Eigenvalues, expansion, conductance, and random walks
 Approximate counting, approximate sampling, and the MCMC method
 Random Spanning trees
 Counting colorings in boundeddegree 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 LeightonRao 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 AroraRaoVazirani relaxation
 Feb 13. Rounding the AroraRaoVazirani relaxation
Feb 18. No class  President's day
 Feb 20. The problem of finding wellseparated sets
 Feb 25. Finding wellseparated sets
 Feb 27. Other results and open problems in spectral graph theory
 Mar 4. GabberGalil expanders
[notes]
midterm and final