135 research outputs found

    Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

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    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 C\mathcal C and show that if p∈Cp\in \cal C and there is a disk DD centered at zero in the complex plane such that p(G)p(G) does not vanish on DD for all bounded degree graphs GG, then for each zz in the interior of DD there exists a deterministic polynomial-time approximation algorithm for evaluating p(G)p(G) at zz. 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

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    We show that if a non-amenable, quasi-transitive, unimodular graph GG 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 GG 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

    Bibliographie

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    Bibliographie

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    Approximate counting using Taylor’s theorem:a survey

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    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

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    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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