6 research outputs found
Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs
For a graph , let be the partition function of the
monomer-dimer system defined by , where is the
number of matchings of size in . We consider graphs of bounded degree
and develop a sublinear-time algorithm for estimating at an
arbitrary value within additive error with high
probability. The query complexity of our algorithm does not depend on the size
of and is polynomial in , and we also provide a lower bound
quadratic in for this problem. This is the first analysis of a
sublinear-time approximation algorithm for a # P-complete problem. Our
approach is based on the correlation decay of the Gibbs distribution associated
with . We show that our algorithm approximates the probability
for a vertex to be covered by a matching, sampled according to this Gibbs
distribution, in a near-optimal sublinear time. We extend our results to
approximate the average size and the entropy of such a matching within an
additive error with high probability, where again the query complexity is
polynomial in and the lower bound is quadratic in .
Our algorithms are simple to implement and of practical use when dealing with
massive datasets. Our results extend to other systems where the correlation
decay is known to hold as for the independent set problem up to the critical
activity
Independent sets, matchings, and occupancy fractions
We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tetali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tre
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Approximate counting, phase transitions and geometry of polynomials
In classical statistical physics, a phase transition is understood by studying the geometry (the zero-set) of an associated polynomial (the partition function). In this thesis, we will show that one can exploit this notion of phase transitions algorithmically, and conversely exploit the analysis of algorithms to understand phase transitions. As applications, we give efficient deterministic approximation algorithms (FPTAS) for counting -colorings, and for computing the partition function of the Ising model