Markov Chain Monte Carlo Methods
Fall 2006, Georgia Tech
Tuesday and Thursday, 9:3011am, in Cherry Emerson room 322
Instructor: Eric Vigoda
Textbook:
I have some lecture notes which I'll post. Also there's a nice
monograph by Mark Jerrum
covering many of the topics in this course.
They are also available on
his webpage,
though the book is cheap.
Homeworks:
For project details
go here
HW 4 pdf: due Thursday October 19
HW 3 pdf: due Tuesday October 3
HW 2 pdf: due Thursday Sept 14
Here's a very rough schedule to give you an idea of the topics we'll cover.
Many of the dates will probably change as we go along.
Lectures:
 Lectures 12 (8/22, 8/24):
Classical Exact Counting Algorithms
 Spanning Trees (Kirchoff's MartrixTree Theorem)
 Kasteleyn's polytime algorithm for the Permanent of Planar graphs
 Lecture notes:
PDF

See Section 1 of Jerrum's book
for a different proof of Kirchoff's result.

Lecture 3 (8/29):
Complexity Class #P, and the Permanent is #Pcomplete

 Lecture notes:
PDF

See Section 2 of Jerrum's book.

Lectures 45 (8/31, 9/5): Counting versus Sampling
 Reductions between Approximate Counting and Approximate Sampling
 Lecture notes:
PDF


See Sections 3.1/3.2 of Jerrum's book.

Lecture 6 (9/7): Sampling: Markov Chain Fundamentals
 Coupling technique
 Ergodic Markov chains have a unique stationary distribution

Lecture notes:
PDF
 HW 2 pdf: due Thursday Sept 14

Lecture 7 (9/12): Coupling from the Past

Lecture notes:
PDF

Lectures 89 (9/14, 9/19): Bounding mixing time via coupling
 Random spanning trees
 Path coupling technique
 Random Colorings

Lecture notes:
PDF

Lectures 10 (9/21): Coupling application: Lozenge tilings
 Lecture by Dana Randall

Lectures 11 (9/26): Linear extensions
 Generating a random linear extension of a partial order
 Notes: see Section 4.3 of Jerrum's book.
 HW 3 pdf: due Tuesday October 3

Lectures 12 (9/28): Advanced coupling
 Random colorings  avoiding the worst case and nonMarkovian couplings
 Notes: see the following survey

Lectures 1314 (10/3, 10/5): Spectral methods
 Canonical Paths
 Generating a random matching

Notes: see Chapter 5 of Jerrum's book.

Lectures 1516 (10/10, 10/12): Approximating the permanent of nonnegative matrices
 HW 4 pdf: due Thursday October 19
 Supplemental notes:
postscript,
PDF, including an
 algorithm/proof sketch
for general bipartite graphs

Lecture 17 (10/19): Ising Model
Connections between phase transitions in Statistical Physics models and
fast covergence of Markov chains
 Strong spatial mixing and O(nlogn) mixing time of the Glauber dynamics

Lecture 18 (10/24): Counting/Sampling Algorithms for the Ising Model
 Approximating the partition function via the hightemperature expansion
 Random sampling via the randomcluster representation

Lecture 19 (10/26):
Conductance
 Bounding the mixing time via conductance

Lecture 20 (10/31): Torpid mixing for the Glauber dynamics
 Contours argument
 Lecture by Dana Randall

Lectures 2123 (11/2, 11/7, 11/9): Estimating the volume of convex bodies
 Lectures by Santosh Vempala
 Notes: see
survey article by Santosh
(also, Section 6 of Jerrum's book)

Lecture 24 (11/14): Approximate Counting via Dynamic Programming
 Dyer's #Knapsack result
 Lecture notes:
PDF


November 20:
Make sure to attend the DIMACS lectures

Week 13 (11/28): Weitz's deterministic approx counting alg for independent sets

Week 14 (11/30, 12/5, 12/7): Project Presentations
 Schedule of talks