CS359G: Graph Partitioning and Expanders

[general info]  [lecture notes] [coursework]

general information

Instructor: Luca Trevisan, Gates 474, Tel. 650 723-8879, email trevisan at stanford dot edu

Classes are Tuesday-Thursday, 2:15-3:30pm, location TBA

Office hours: TBA

About the course

The mathematics of expander graphs is studied by three distinct communities:
  1. 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.
  2. 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.
  3. 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: a basic course in linear algebra and a course on algorithms; preferably, also a basic understanding of linear programming and of duality.

Assignments: two 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.


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:
  1. Definitions: edge and vertex expansion, uniform and general sparsest cut problems, review of linear algebra
  2. Eigenvalues and expansion, Cheeger's inequality and the spectral partitioning algorithm
  3. Cheeger's inequality, continued
  4. Classes of graphs for which spectral partitioning is provably good
  5. Eigenvalues, expansion, conductance, and random walks
  6. Applications of expanders: derandomization
  7. Applications of expanders: security amplification of one-way permutations
  8. The Margulis-Gabor-Galil construction of expanders
  9. The Zig-Zag graph product construction
  10. Algorithms for finding sparse cuts: Leighton-Rao, and metric embeddings
  11. Equivalence of rounding the Leighton-Rao relaxation and embedding general metrics into L1
  12. Algorithms for finding sparse cuts: Arora-Rao-Vazirani
  13. Arora-Rao-Vazirani, continued
  14. Integrality gaps for the Arora-Rao-Vazirani relaxation
  15. Approximate counting, approximate sampling, and the MCMC method
  16. Random Spanning trees
  17. Counting colorings in bounded-degree graphs
  18. Counting perfect matchings in dense bipartite graphs
  19. The Metropolis algorithm