118 research outputs found
Fast Algorithms for General Spin Systems on Bipartite Expanders
A spin system is a framework in which the vertices of a graph are assigned
spins from a finite set. The interactions between neighbouring spins give rise
to weights, so a spin assignment can also be viewed as a weighted graph
homomorphism. The problem of approximating the partition function (the
aggregate weight of spin assignments) or of sampling from the resulting
probability distribution is typically intractable for general graphs.
In this work, we consider arbitrary spin systems on bipartite expander
-regular graphs, including the canonical class of bipartite random
-regular graphs. We develop fast approximate sampling and counting
algorithms for general spin systems whenever the degree and the spectral gap of
the graph are sufficiently large. Our approach generalises the techniques of
Jenseen et al. and Chen et al. by showing that typical configurations on
bipartite expanders correspond to "bicliques" of the spin system; then, using
suitable polymer models, we show how to sample such configurations and
approximate the partition function in time, where is the
size of the graph
Algorithms for the ferromagnetic Potts model on expanders
We give algorithms for approximating the partition function of the
ferromagnetic Potts model on -regular expanding graphs. We require much
weaker expansion than in previous works; for example, the expansion exhibited
by the hypercube suffices. The main improvements come from a significantly
sharper analysis of standard polymer models, using extremal graph theory and
applications of Karger's algorithm to counting cuts that may be of independent
interest. It is #BIS-hard to approximate the partition function at low
temperatures on bounded-degree graphs, so our algorithm can be seen as evidence
that hard instances of #BIS are rare. We believe that these methods can shed
more light on other important problems such as sub-exponential algorithms for
approximate counting problems.Comment: 27 page
Fast and Perfect Sampling of Subgraphs and Polymer Systems
We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications
Counting in two-spin models on d-regular graphs
We establish that the normalized log-partition function of any two-spin
system on bipartite locally tree-like graphs converges to a limiting “free
energy density” which coincides with the (nonrigorous) Bethe prediction of
statistical physics. Using this result, we characterize the local structure of
two-spin systems on locally tree-like bipartite expander graphs without the
use of the second moment method employed in previous works on these questions.
As a consequence, we show that for both the hard-core model and the
anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to
approximate the partition function or approximately sample from the model
on d-regular graphs when the model has nonuniqueness on the d-regular tree.
Together with results of Jerrum–Sinclair, Weitz, and Sinclair–Srivastava–
Thurley, this gives an almost complete classification of the computational
complexity of homogeneous two-spin systems on bounded-degree graphs.Supported in part by Alfred P. Sloan Research Fellowship. Supported in part by Department of Defense NDSEG Fellowships
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
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