114 research outputs found
Fast Algorithms at Low Temperatures via Markov Chains
For spin systems, such as the hard-core model on independent sets weighted by fugacity lambda>0, efficient algorithms for the associated approximate counting/sampling problems typically apply in the high-temperature region, corresponding to low fugacity. Recent work of Jenssen, Keevash and Perkins (2019) yields an FPTAS for approximating the partition function (and an efficient sampling algorithm) on bounded-degree (bipartite) expander graphs for the hard-core model at sufficiently high fugacity, and also the ferromagnetic Potts model at sufficiently low temperatures. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. We present a simple discrete-time Markov chain for abstract polymer models, and present an elementary proof of rapid mixing of this new chain under sufficient decay of the polymer weights. Applying these general polymer results to the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs yields fast algorithms with running time O(n log n) for the Potts model and O(n^2 log n) for the hard-core model, in contrast to typical running times of n^{O(log Delta)} for algorithms based on Barvinok\u27s polynomial interpolation method on graphs of maximum degree Delta. In addition, our approach via our polymer model Markov chain is conceptually simpler as it circumvents the zero-free analysis and the generalization to complex parameters. Finally, we combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin system Glauber dynamics restricted to even and odd or "red" dominant portions of the respective state spaces
Algorithmic Pirogov-Sinai theory
We develop an efficient algorithmic approach for approximate counting and
sampling in the low-temperature regime of a broad class of statistical physics
models on finite subsets of the lattice and on the torus
. Our approach is based on combining contour
representations from Pirogov-Sinai theory with Barvinok's approach to
approximate counting using truncated Taylor series. Some consequences of our
main results include an FPTAS for approximating the partition function of the
hard-core model at sufficiently high fugacity on subsets of with
appropriate boundary conditions and an efficient sampling algorithm for the
ferromagnetic Potts model on the discrete torus at
sufficiently low temperature
Algorithms for #BIS-hard problems on expander graphs
We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) Δ-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite Δ-regular tree
Algorithms for #BIS-hard problems on expander graphs
We give an FPTAS and an efficient sampling algorithm for the high-fugacity
hard-core model on bounded-degree bipartite expander graphs and the
low-temperature ferromagnetic Potts model on bounded-degree expander graphs.
The results apply, for example, to random (bipartite) -regular graphs,
for which no efficient algorithms were known for these problems (with the
exception of the Ising model) in the non-uniqueness regime of the infinite
-regular tree. We also find efficient counting and sampling algorithms
for proper -colorings of random -regular bipartite graphs when
is sufficiently small as a function of
Sampling from the low temperature Potts model through a Markov chain on flows
In this paper we consider the algorithmic problem of sampling from the Potts
model and computing its partition function at low temperatures. Instead of
directly working with spin configurations, we consider the equivalent problem
of sampling flows. We show, using path coupling, that a simple and natural
Markov chain on the set of flows is rapidly mixing. As a result we find a
-approximate sampling algorithm for the Potts model at low enough
temperatures, whose running time is bounded by for
graphs with edges.Comment: Slightly revised version based on referee comments. No significant
changes. Accepted in Random Structures and Algorithm
- …