Graph Partitioning and Expanders
A summer course to be offered at the University of Salerno on July 12-20.
About the course
The mathematics of expander graphs is studied by three distinct
In this course we aim to present key results from these three areas, and
to explore the common mathematical background.
- 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
Prerequisites: a basic course in linear algebra and a course on
algorithms; preferably, also a basic understanding of linear programming
and of duality
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.
There will be 14 lectures, of 90 minutes each.
The following is the initial tentative schedule:
- Day one, Lectures 1-2.
- Definition of expansion and sparsest cut
- Overview of the three parts of the course
- Introduction to spectral graph theory
- Easy direction of Cheeger's inequality
- Day two, Lectures 3-4.
- Cheeger's inequality
- Cayley graphs
- Characters of abelian groups
- Day three, Lectures 5-6.
- Characters of abelian groups (continued)
- Spectra of the cycle, the hypercube and the clique
- The Leighton-Rao relaxation of sparsest cut
- Rounding the LR relaxation using embeddings in L1
- Day four, Lectures 7-8.
- Proof of Bourgain's theorem on embedding general metrics in L1.
- Introduction to semidefinite programming.
- Day five, Lectures 9-10.
- More on semidefinite matrices
- The ARV relaxation of sparsest cut and of balanced separator
- Rounding the ARV relaxation of balanced separator (brief sketch)
- Definitions: edge and vertex expansion, uniform and general sparsest
cut problems, review of linear algebra
- Eigenvalues and expansion, Cheeger's inequality and the spectral
- Cheeger's inequality, continued
- Eigenvalues, expansion, conductance, and random walks
- 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
- Algorithms for finding sparse cuts: Leighton-Rao, and metric embeddings
- Algorithms for finding sparse cuts: Arora-Rao-Vazirani
- Arora-Rao-Vazirani, continued
- Approximate counting, approximate sampling, and the MCMC method
- Counting colorings in bounded-degree graphs
- Counting perfect matchings in dense graphs