## CS294-180: PARTITION FUNCTIONS: ALGORITHMS & COMPLEXITY, FALL 2020

INSTRUCTOR: Alistair Sinclair
(sinclair@cs)

TIME: Tuesday, Thursday 9:30-11:00

PLACE: Online (Zoom)

OFFICE HOURS: Monday 1:00-2:00, Tuesday 11:00-12:00 Online

## RECENT ANNOUNCEMENTS

(9/15/20) The note for Lectures 5 and 6 have been posted below. Please notify me
of any errors or typos you find.

(9/8/20) The note for Lecture 4 has been posted below.

(9/3/20) Please recall that the lectures and Piazza forum are open only to
Berkeley students enrolled in the class, or others by permission. Please contact
me if you are not able to enroll. If you wish to audit the class, please enroll S/U.

(9/3/20) The third lecture note has been posted below. You are strongly encouraged
to read the notes and do the exercises after each lecture..

(9/1/20) The second lecture note has been posted below. (The second half of the
note will be covered at the start of the next lecture.) You are strongly encouraged
to read the notes after each lecture and check the details (some of which are pointed
out as exercises in the text). Again, please let me know about any
typos or errors.

(8/27/20) The first lecture note has been posted below. (The second half of the
note will be covered at the start of the next lecture.) Please let me know about any
typos, errors or explanations that can be improved in the notes and I'll be happy to
fix them. Please also be sure to read the
enrollment policies in today's post (#7) on Piazza.

(8/3/20) Welcome to the course! Please be sure to sign up for the class
Piazza forum at https://piazza.com/berkeley/fall2020/cs294180, where you can ask
questions about the class and where
Zoom links for lectures and office hours will be posted.

## COURSE DESCRIPTION

Partition functions are a class of polynomials with combinatorial coefficients that count
weighted combinatorial structures. Originally introduced in statistical physics, they have
more recently found widespread application in combinatorics, theoretical computer science,
statistics and machine learning, and have become a vibrant research area in their own
right. This course will cover algorithmic and complexity theoretic aspects of
partition functions,
as well as their connections to physical and combinatorial phase transitions.
A large portion of the class will be devoted to the Markov chain Monte Carlo method, which
remains the most widely applicable algorithmic tool in the area. The course will also cover
more recent deterministic algorithms based on correlation decay and geometry of polynomials.
The class is open to graduate students and exceptionally well-prepared undergraduates
(with the permission of the instructor). Prerequisites include a solid background in algorithms
and complexity theory and appropriate mathematical sophistication.

**Tentative List of Topics**

Counting, partition functions and Markov random fields.
The complexity class #P and #P-completeness. Approximate counting and
approximating partition functions. Relationship between counting and random sampling.
Polynomial time examples: spanning trees and matchings in planar graphs.
Basis of the Markov chain Monte Carlo (MCMC) method: ergodicity,
stationary distributions, reversibility, mixing times.
Coupling of Markov chains; applications to graph colorings and statistical
physics. Dobrushin uniqueness condition.
Functional analysis for mixing times: spectral gap and log-Sobolev inequalities.
Connection to multicommodity flows. Applications to approximating the permanent,
and counting the bases of a matroid.
Lower bounds: conductance and isoperimetric inequalities.
Correlation decay and spatial mixing in statistical physics and MRFs;
deterministic approximation algorithms; application to independent sets
(hard-core model).
Hardness of approximation based on non-uniqueness of the Gibbs measure.
Geometry of polynomials: approximation algorithms based on absence of
complex zeros. Lee-Yang theorems and connection to phase transitions.
Applications to the Ising model, graph colorings etc.

## MATERIALS

There is no textbook for the class; most of the material is drawn from research
papers, and references will be given as we go along. The primary written resource for the
class will be detailed lecture notes, which will be posted on this web page shortly after each
class. I will provide many of these, but will ask students to take turns at scribing
some of the lectures as necessary (working individually or in pairs).

## ASSESSMENT/ACTIVITIES

The main requirement of the class is following the lecture material via
Zoom and lecture notes.
Assessment will be based on some combination of (i) an end-of-semester
project (involving presentation of and/or incremental work on a
research paper); (ii) homeworks; (iii) scribe notes; and (iv) Zoom classroom
attendance/participation.
The exact combination will depend on the enrollment and
composition of the class. Further details will be circulated
early in the semester. Workload (beyond keeping up with the material)
is not expected to be particularly heavy.

## LECTURE NOTES