CS294180 Partition Functions: Project Suggestions
Below is a list of suggestions for project topics. Please read the
detailed guidelines here first.
You are also free to propose your own topic, provided you send me details
and get it approved first.
Once you have made your selections, please email me your first choice
plus at least one, and preferably two, alternatives.
Projects marked XXX have already been selected by someone.
Deadline for selecting a project: Monday November 2nd!
Note: Most of the suggestions contain multiple papers. These should be
viewed as a guide, and in most cases are maximal for the scope. In most
cases you will probably decide to focus mainly on one of the papers, or
on parts of some of them.
Please discuss your proposed scope with me before finalizing your presentation
(but after you've got your topic).

HammersleyClifford for hard constraints. [Moussouris, Journal of Statistical Physics,
1974; Gandolfi & Lenarda, Mathematics & Mechanics of Complex Systems, 2016.]
This is intended as a more openended project: find out what is known (in addition to
these papers) and conduct a critical review of it.

An approximation algorithm for network reliability.
[Guo & Jerrum SIAM Journal on Computing, 2019; Guo, Jerrum & Liu,
ACM STOC, 2017.]
Complements Karger's algorithm from Lecture 5; the algorithm builds on the
earlier paper, which is related to the algorithmic Lovasz Local Lemma.

Group representations and mixing times. [Diaconis & Shashahani,
Zeitschrift fuer Wahrscheinlichkeitstheorie, 1981;
Diaconis, Group Representations in Probability & Statistics, 1988.]
For those interested in group theory; explore the use of group representations to
analyze mixing times of cardshuffling and other Markov chains.

Generating random spanning trees. [Aldous, SIAM Journal of Discrete
Math, 1990; Broder, IEEE FOCS, 1989; Wilson, ACM STOC, 1996;
Madry, Straszak & Tarnawski, ACMSIAM SODA, 2015; Schild, ACM STOC, 2018;
Anari, Liu, Oveis Gharan & Vinzant, arXiv 2004.07220, 2020.]
Various approaches, of increasing sophistication,
to the important problem of generating random spanning trees (faster than the
obvious reduction to counting via the matrix tree theorem).

The Bipartite Independent Set problem (BIS).
[Dyer, Goldberg, Greenhill & Jerrum, Algorithmica, 2003; Goldberg & Jerrum,
JACM, 2012.]
A class of approximate counting problems, all equivalent to counting independent sets
in bipartite graphs, whose status is still unresolved.

More on coupling for colorings. [Vigoda, IEEE FOCS, 1999;
Hayes/Vigoda, ACMSIAM SODA, 2005; Chen, Delcourt, Moitra, Perarnau & Postle,
ACMSIAM SODA, 2019.]
Vigoda's 11/6 Δ algorithm and a recent improvement, plus
better algorithms with stronger assumptions.

Coupling from the Past. [Propp & Wilson, Random Structures & Algorithms, 1996;
Fill, Annals of Applied Prob., 1998; Fill, Machida, Murdoch & Rosenthal,
Random Structures & Algorithms, 2000.]
A twist on coupling that eliminates all bias from the samples.

Mixing time of the Thorp shuffle. [Morris, Annals of Probability, 2009;
Morris, Combinatorics, Probability & Computing, 2013.]
A delicate use of entropy arguments to solve a notorious problem about card shuffling.

Approximating the volume of a convex body. [Lovasz & Vempala, IEEE FOCS, 2003;
Cousins & Vempala, ACM STOC, 2015; Cousins & Vempala, Math Programming C,
2016.]
Another fundamental application of MCMC that we didn't have time for in the class;
the latter papers are the state of the art.

The Ising model on trees. [Kenyon, Mossel & Peres, IEEE FOCS, 2001;
Martinelli, Sinclair & Weitz, IEEE FOCS, 2003.]
Detailed analysis of Glauber dynamics for the Ising model on trees, with connections
to phase transitions.

Matroid bases via highdimensional expanders.
[Anari, Liu, Oveis Gharan & Vinzant, IEEE FOCS, 2018;
Alev & Lau, ACM STOC, 2020.]
The original paper on this result, which replaces the logSobolev approach of Cryan
et al. we saw in class by a spectral analysis. The second paper simplifies and
generalizes the approach.

Correlation decay for colorings.
[Garmarnik & Katz, Journal of Discrete Algorithms, 2012;
Gamarnik, Katz & Misra, Random Structures & Algorithms, 2015.]
Attempts to extend the Weitz SAW algorithm beyond twospin systems, with
limited success.

Inapproximability of antiferromagnetic twospin systems in the nonuniqueness region.
[Sly & Sun, Annals of Probability, 2014; Galanis, Stefankovic & Vigoda,
Combinatorics, Probability & Computing , 2016; Galanis, Stefankovic & Vigoda
JACM, 2015.]
Two alternative approaches to proving hardness of
approximation for the partition function of the antiferromagnetic Ising model
in the nonuniqueness region, generalizing Sly's original result, and an
extension to multispin systems. These papers are quite technical; any one of
them would suffice.

Correlation decay for hard spheres.
[Helmuth, Perkins & Petti, ArXiv, January 2020.]
This was the subject of the talk by Samantha Petti at the Simons Institute workshop
we attended as a virtual field trip in September.

Approximate counting via the cluster expansion.
[Helmuth, Perkins & Regts, Probability Theory & Related Fields, 2020.]
Use of the cluster expansion from statistical physics to obtain algorithms
for spin systems at low temperatures.
 Connections to quantum computing.
[Ji, Jin & Lu, ACMSIAM SODA, 2021;
Harrow, Mehraban & Soleimanifar, ACM STOC, 2020.]
Two recent examples where zeros of partition functions play a role in quantum
computing.