38 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

    Approximating the Permanent of a Random Matrix with Vanishing Mean

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    We show an algorithm for computing the permanent of a random matrix with vanishing mean in quasi-polynomial time. Among special cases are the Gaussian, and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time 2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the intuition that the permanent is hard because of the "sign problem" - namely the interference between entries of a matrix with different signs. A major open question then remains whether one can provide an efficient algorithm for random matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the baseline assumptions of the BosonSampling paradigm

    Weighted counting of solutions to sparse systems of equations

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    Given complex numbers w1,…,wnw_1, \ldots, w_n, we define the weight w(X)w(X) of a set XX of 0-1 vectors as the sum of w1x1β‹―wnxnw_1^{x_1} \cdots w_n^{x_n} over all vectors (x1,…,xn)(x_1, \ldots, x_n) in XX. We present an algorithm, which for a set XX defined by a system of homogeneous linear equations with at most rr variables per equation and at most cc equations per variable, computes w(X)w(X) within relative error Ο΅>0\epsilon >0 in (rc)O(ln⁑nβˆ’ln⁑ϡ)(rc)^{O(\ln n-\ln \epsilon)} time provided ∣wjβˆ£β‰€Ξ²(rc)βˆ’1|w_j| \leq \beta (r \sqrt{c})^{-1} for an absolute constant Ξ²>0\beta >0 and all j=1,…,nj=1, \ldots, n. A similar algorithm is constructed for computing the weight of a linear code over Fp{\Bbb F}_p. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.Comment: The exposition is improved, a couple of inaccuracies correcte
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