141 research outputs found
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
In this paper we show a new way of constructing deterministic polynomial-time
approximation algorithms for computing complex-valued evaluations of a large
class of graph polynomials on bounded degree graphs. In particular, our
approach works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
Factor-of-iid balanced orientation of non-amenable graphs
We show that if a non-amenable, quasi-transitive, unimodular graph has
all degrees even then it has a factor-of-iid balanced orientation, meaning each
vertex has equal in- and outdegree. This result involves extending earlier
spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of
quasi-transitive, unimodular graphs. As a consequence, we also obtain that when
is regular (of either odd or even degree) and bipartite, it has a
factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov
beyond transitive graphs.Comment: 24 pages, 1 figure. This is one of two papers that are replacing the
shorter arXiv submission arXiv:2101.12577v1 Factor of iid Schreier decoration
of transitive graph
Approximate counting using Taylor’s theorem:a survey
In this article we consider certain well-known polynomials associated with graphs including the independence polynomial and the chromatic polynomial. These polynomials count certain objects in graphs: independent sets in the case of the independence polynomial and proper colourings in the case of the chro- matic polynomial. They also have interpretations as partition functions in statistical physics.The algorithmic problem of (approximately) computing these types of polyno- mials has been studied for close to 50 years, especially using Markov chain tech- niques. Around eight years ago, Barvinok devised a new algorithmic approach based on Taylor’s theorem for computing the permanent of certain matrices, and the approach has been applied to various graph polynomials since then. This arti- cle is intended as a gentle introduction to the approach as well as a partial survey of associated techniques and results
On the Hill relation and the mean reaction time for metastable processes
We illustrate how the Hill relation and the notion of quasi-stationary
distribution can be used to analyse the error introduced by many algorithms
that have been proposed in the literature, in particular in molecular dynamics,
to compute mean reaction times between metastable states for Markov processes.
The theoretical findings are illustrated on various examples demonstrating the
sharpness of the error analysis as well as the applicability of our study to
elliptic diffusions
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