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.

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 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