CS294-180 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. Note that such proposals should be broadly
in line with the themes and spirit of this class.
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: Tuesday October 31st!
Note: Most of the suggestions contain multiple papers. These should be
viewed as a guide, and are maximal for the scope. Most likely you will
end up covering only one of the papers, and/or parts of the papers.
Please discuss your proposed scope with me before finalizing your presentation
(but after you've got your topic).
-
Hammersley-Clifford for hard constraints. [Moussouris, Journal of Statistical Physics,
1974; Gandolfi & Lenarda, Mathematics & Mechanics of Complex Systems, 2016.]
This is intended as an open-ended project: find out what is known (in addition to
these papers), conduct a critical review of it, and ideally propose new approaches.
-
Group representations and mixing times. [Diaconis & Shashahani,
Zeitschrift fuer Wahrscheinlichkeitstheorie, 1981;
Diaconis, Group Representations in Probability & Statistics, 1988.]
For those with a strong interest in group theory; explore the use of group representations to
analyze mixing times of card-shuffling and other Markov chains.
-
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 2017 paper, which is related to the algorithmic Lovasz Local Lemma.
-
Generating random spanning trees. [Aldous, SIAM Journal of Discrete
Math, 1990; Broder, IEEE FOCS, 1989; Wilson, ACM STOC, 1996;
Madry, Straszak & Tarnawski, ACM-SIAM SODA, 2015; Anari, Liu & Vuong,
IEEE FOCS, 2022.]
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 last paper
makes use of entropy methods.
-
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, ACM-SIAM SODA, 2005; Chen, Delcourt, Moitra, Perarnau & Postle,
ACM-SIAM SODA, 2019.]
Vigoda's 11/6 Δ algorithm and a recent improvement, plus
better algorithms with stronger assumptions.
-
XXX An almost-optimal algorithm for colorings.
[Chen, Liu, Mani & Moitra, arXiv 2304.01954]
This very recent breakthrough proves rapid mixing (as well as strong spatial mixing)
for the colorings dynamics all the way down to Δ + 3 colors.
-
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.
-
XXX 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.
-
Matroid bases via high-dimensional expanders.
[Anari, Liu, Oveis Gharan & Vinzant, IEEE FOCS, 2018;
Alev & Lau, ACM STOC, 2020.]
The original paper on this result, which replaces the log-Sobolev approach of Cryan
et al. we saw in class by a spectral analysis. The second paper simplifies and
generalizes the approach.
-
Optimal mixing times for Glauber dynamics.
[Chen, Liu & Vigoda, ACM STOC, 2021; Anari, Jain, Koehler, Pham & Vuong,
ACM STOC, 2022.]
Two of the latest in a long line of papers improving the mixing time of Glauber dynamics
to O(n log n) in almost all cases where polynomial mixing time is known, via a sophisticated
use of entropy methods.
-
Inapproximability of antiferromagnetic two-spin systems in the non-uniqueness 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 anti-ferromagnetic Ising model
in the non-uniqueness region, generalizing Sly's original result, and an
extension to multi-spin systems.
-
XXX 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.
-
XXX Connections to quantum computing.
[Ji, Jin & Lu, ACM-SIAM SODA, 2021;
Harrow, Mehraban & Soleimanifar, ACM STOC, 2020.]
Two recent examples where zeros of partition functions play a role in quantum
computing.
-
XXX The cutoff phenomenon for mixing times.
[Levin, Luczak & Peres, Probability Theory & Related Fields, 2010.]
A nice example (Ising model on the complete graph) that exhibits a
cutoff phenomenon and can be cleanly analyzed. (There are many other
interesting papers on cutoff.)
-
XXX Kawasaki dynamics for the Ising model.
[Bauershmidt, Bodineau & Dagallier, arXiv 2023.]
Application of modified log-Sobolev inequalities to prove rapid mixing
of Kawasaki dynamics below the uniqueness threshold for the Ising model
on random graphs.