Markov Chain Monte Carlo Methods

Fall 2006, Georgia Tech
Tuesday and Thursday, 9:30-11am, 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.

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 1-2 (8/22, 8/24): Classical Exact Counting Algorithms
Spanning Trees (Kirchoff's Martrix-Tree Theorem)
Kasteleyn's poly-time 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 #P-complete
Lecture notes: PDF
See Section 2 of Jerrum's book.

Lectures 4-5 (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 8-9 (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 non-Markovian couplings
Notes: see the following survey

Lectures 13-14 (10/3, 10/5): Spectral methods
Canonical Paths
Generating a random matching
Notes: see Chapter 5 of Jerrum's book.

Lectures 15-16 (10/10, 10/12): Approximating the permanent of non-negative 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 high-temperature expansion
Random sampling via the random-cluster 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 21-23 (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