168 research outputs found
Maximum Matchings via Glauber Dynamics
In this paper we study the classic problem of computing a maximum cardinality
matching in general graphs . The best known algorithm for this
problem till date runs in time due to Micali and Vazirani
\cite{MV80}. Even for general bipartite graphs this is the best known running
time (the algorithm of Karp and Hopcroft \cite{HK73} also achieves this bound).
For regular bipartite graphs one can achieve an time algorithm which,
following a series of papers, has been recently improved to by
Goel, Kapralov and Khanna (STOC 2010) \cite{GKK10}. In this paper we present a
randomized algorithm based on the Markov Chain Monte Carlo paradigm which runs
in time, thereby obtaining a significant improvement over
\cite{MV80}.
We use a Markov chain similar to the \emph{hard-core model} for Glauber
Dynamics with \emph{fugacity} parameter , which is used to sample
independent sets in a graph from the Gibbs Distribution \cite{V99}, to design a
faster algorithm for finding maximum matchings in general graphs. Our result
crucially relies on the fact that the mixing time of our Markov Chain is
independent of , a significant deviation from the recent series of
works \cite{GGSVY11,MWW09, RSVVY10, S10, W06} which achieve computational
transition (for estimating the partition function) on a threshold value of
. As a result we are able to design a randomized algorithm which runs
in time that provides a major improvement over the running time
of the algorithm due to Micali and Vazirani. Using the conductance bound, we
also prove that mixing takes time where is the size
of the maximum matching.Comment: It has been pointed to us independently by Yuval Peres, Jonah
Sherman, Piyush Srivastava and other anonymous reviewers that the coupling
used in this paper doesn't have the right marginals because of which the
mixing time bound doesn't hold, and also the main result presented in the
paper. We thank them for reading the paper with interest and promptly
pointing out this mistak
How quickly can we sample a uniform domino tiling of the 2L x 2L square via Glauber dynamics?
TThe prototypical problem we study here is the following. Given a square, there are approximately ways to tile it with
dominos, i.e. with horizontal or vertical rectangles, where
is Catalan's constant [Kasteleyn '61, Temperley-Fisher '61]. A
conceptually simple (even if computationally not the most efficient) way of
sampling uniformly one among so many tilings is to introduce a Markov Chain
algorithm (Glauber dynamics) where, with rate , two adjacent horizontal
dominos are flipped to vertical dominos, or vice-versa. The unique invariant
measure is the uniform one and a classical question [Wilson
2004,Luby-Randall-Sinclair 2001] is to estimate the time it takes to
approach equilibrium (i.e. the running time of the algorithm). In
[Luby-Randall-Sinclair 2001, Randall-Tetali 2000], fast mixin was proven:
for some finite . Here, we go much beyond and show that . Our result applies to rather general domain
shapes (not just the square), provided that the typical height
function associated to the tiling is macroscopically planar in the large
limit, under the uniform measure (this is the case for instance for the
Temperley-type boundary conditions considered in [Kenyon 2000]). Also, our
method extends to some other types of tilings of the plane, for instance the
tilings associated to dimer coverings of the hexagon or square-hexagon
lattices.Comment: to appear on PTRF; 42 pages, 9 figures; v2: typos corrected,
references adde
Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion
We prove an optimal mixing time bound on the single-site update Markov chain
known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our
work presents an improved version of the spectral independence approach of
Anari et al. (2020) and shows mixing time on any -vertex graph
of bounded degree when the maximum eigenvalue of an associated influence matrix
is bounded. As an application of our results, for the hard-core model on
independent sets weighted by a fugacity , we establish
mixing time for the Glauber dynamics on any -vertex graph of constant
maximum degree when where
is the critical point for the uniqueness/non-uniqueness
phase transition on the -regular tree. More generally, for any
antiferromagnetic 2-spin system we prove mixing time of the
Glauber dynamics on any bounded degree graph in the corresponding tree
uniqueness region. Our results apply more broadly; for example, we also obtain
mixing for -colorings of triangle-free graphs of maximum
degree when the number of colors satisfies where
, and mixing for generating random
matchings of any graph with bounded degree and edges
Fast sampling via spectral independence beyond bounded-degree graphs
Spectral independence is a recently-developed framework for obtaining sharp
bounds on the convergence time of the classical Glauber dynamics. This new
framework has yielded optimal sampling algorithms on
bounded-degree graphs for a large class of problems throughout the so-called
uniqueness regime, including, for example, the problems of sampling independent
sets, matchings, and Ising-model configurations.
Our main contribution is to relax the bounded-degree assumption that has so
far been important in establishing and applying spectral independence. Previous
methods for avoiding degree bounds rely on using -norms to analyse
contraction on graphs with bounded connective constant (Sinclair, Srivastava,
Yin; FOCS'13). The non-linearity of -norms is an obstacle to applying
these results to bound spectral independence. Our solution is to capture the
-analysis recursively by amortising over the subtrees of the recurrence
used to analyse contraction. Our method generalises previous analyses that
applied only to bounded-degree graphs.
As a main application of our techniques, we consider the random graph
, where the previously known algorithms run in time
or applied only to large . We refine these algorithmic bounds significantly,
and develop fast algorithms based on Glauber dynamics that apply
to all , throughout the uniqueness regime
Random sampling of lattice configurations using local Markov chains
Algorithms based on Markov chains are ubiquitous across scientific disciplines, as they provide a method for extracting statistical information about large, complicated systems. Although these algorithms may be applied to arbitrary graphs, many physical applications are more naturally studied under the restriction to regular lattices. We study several local Markov chains on lattices, exploring how small changes to some parameters can greatly influence efficiency of the algorithms.
We begin by examining a natural Markov Chain that arises in the context of "monotonic surfaces", where some point on a surface is sightly raised or lowered each step, but with a greater rate of raising than lowering. We show that this chain is rapidly mixing (converges quickly to the equilibrium) using a coupling argument; the novelty of our proof is that it requires defining an exponentially increasing distance function on pairs of surfaces, allowing us to derive near optimal results in many settings.
Next, we present new methods for lower bounding the time local chains may take to converge to equilibrium. For many models that we study, there seems to be a phase transition as a parameter is changed, so that the chain is rapidly mixing above a critical point and slow mixing below it. Unfortunately, it is not always possible to make this intuition rigorous. We present the first proofs of slow mixing for three sampling problems motivated by statistical physics and nanotechnology: independent sets on the triangular lattice (the hard-core lattice gas model), weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model), and non-saturated Ising (tile-based self-assembly). Previous proofs of slow mixing for other models have been based on contour arguments that allow us prove that a bottleneck in the state space constricts the mixing. The standard contour arguments do not seem to apply to these problems, so we modify this approach by introducing the notion of "fat contours" that can have nontrivial area. We use these to prove that the local chains defined for these models are slow mixing.
Finally, we study another important issue that arises in the context of phase transitions in physical systems, namely how the boundary of a lattice can affect the efficiency of the Markov chain. We examine a local chain on the perfect and near-perfect matchings of the square-octagon lattice, and show for one boundary condition the chain will mix in polynomial time, while for another it will mix exponentially slowly. Strikingly, the two boundary conditions only differ at four vertices. These are the first rigorous proofs of such a phenomenon on lattice graphs.Ph.D.Committee Chair: Randall, Dana; Committee Member: Heitsch, Christine; Committee Member: Mihail, Milena; Committee Member: Trotter, Tom; Committee Member: Vigoda, Eri
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