3,103 research outputs found

    Modified Bethe Permanent of a Nonnegative Matrix

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    Currently the best deterministic polynomial-time algorithm for approximating the permanent of a non-negative matrix is based on minimizing the Bethe free energy function of a certain normal factor graph (NFG). In order to improve the approximation guarantee, we propose a modified NFG with fewer cycles, but still manageable function-node complexity; we call the approximation obtained by minimizing the function of the modified normal factor graph the modified Bethe permanent. For nonnegative matrices of size 3× 3, we give a tight characterization of the modified Bethe permanent. For non-negative matrices of size n× n with n≥ 3, we present a partial characterization, along with promising numerical results. The analysis of the modified NFG is also interesting because of its tight connection to an NFG that is used for approximating a permanent-like quantity in quantum information processing. © 2020 IEEE

    Approximating the Permanent with Fractional Belief Propagation

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    We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the Belief Propagation (BP) approach and its Fractional Belief Propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. The Fractional Free Energy (FFE) functional is parameterized by a scalar parameter γ[1;1]\gamma\in[-1;1], where γ=1\gamma=-1 corresponds to the BP limit and γ=1\gamma=1 corresponds to the exclusion principle (but ignoring perfect matching constraints) Mean-Field (MF) limit. FFE shows monotonicity and continuity with respect to γ\gamma. For every non-negative matrix, we define its special value γ[1;0]\gamma_*\in[-1;0] to be the γ\gamma for which the minimum of the γ\gamma-parameterized FFE functional is equal to the permanent of the matrix, where the lower and upper bounds of the γ\gamma-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of γ\gamma_* varies for different ensembles but γ\gamma_* always lies within the [1;1/2][-1;-1/2] interval. Moreover, for all ensembles considered the behavior of γ\gamma_* is highly distinctive, offering an emprirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.Comment: 42 pages, 14 figure

    Bounds on the permanent and some applications

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    We give new lower and upper bounds on the permanent of a doubly stochastic matrix. Combined with previous work, this improves on the deterministic approximation factor for the permanent. We also give a combinatorial application of the lower bound, proving S. Friedland's "Asymptotic Lower Matching Conjecture" for the monomer-dimer problem

    Degree-MM Bethe and Sinkhorn Permanent Based Bounds on the Permanent of a Non-negative Matrix

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    The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy functions associated with some suitably defined factor graph; the resulting approximation to the permanent is called the Bethe permanent. Vontobel gave a combinatorial characterization of the Bethe permanent via degree-MM Bethe permanents, which is based on degree-MM covers of the underlying factor graph. In this paper, we prove a degree-MM-Bethe-permanent-based lower bound on the permanent of a non-negative matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf. Theory, Mar. 2013]. We also prove a degree-MM-Bethe-permanent-based upper bound on the permanent of a non-negative matrix. In the limit MM \to \infty, these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative matrix. Moreover, we prove similar results for an approximation to the permanent known as the (scaled) Sinkhorn permanent.Comment: submitte

    The number of matchings in random graphs

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    We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the computation of the entropy, i.e. the leading order of the logarithm of the number of solutions, of matchings with a given size. We derive both an algorithm to compute this entropy for an arbitrary graph with a girth that diverges in the large size limit, and an analytic result for the entropy in regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical Mechanic
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