|
VERY TENTATIVE
LECTURE SCHEDULE
Lectures:
- Week 1 (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
|