Fractionally log-concave and sector-stable polynomials: counting planar matchings and more
Fractionally log-concave and sector-stable polynomials: counting planar matchings and more
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DOI：
10.1145/3406325.3451123
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发表时间：
2021-02
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通讯作者：
Yeganeh Alimohammadi;Nima Anari;Kirankumar Shiragur;T. Vuong
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作者：
Yeganeh Alimohammadi;Nima Anari;Kirankumar Shiragur;T. Vuong
我们展示了用于计数给定尺寸的匹配或更一般的平面上采样/计数月二聚体系统的完全多项式近似方案（FPRA），而不是不必要的，而可以准确计数平面图上的完美匹配。在多项式时间内，Jerrum（J Stat Phys 1987）表明了不完美的匹配为＃P-Hard，他也提出了一个问题，即有效近似计数是否为可能。作为我们结果的进一步应用，我们可以通过分区限制的多个位点Glauber动态来表明如何有效地采样非对称确定点过程。实体稳定性和对数cove性，但与它们在分布的有用转换下具有鲁棒性不同。 Anari等人引入的分布和光谱独立性的概念（2020年），为基于多项式的几何形状建立光谱独立性提供了一种新的工具。复杂的平面必须满足我们所谓的分数对数con弹性；这概括了Gårding建立的经典结果在半平面中没有根的多项式必须在正骨上进行对数。
We show fully polynomial time randomized approximation schemes (FPRAS) for counting matchings of a given size, or more generally sampling/counting monomer-dimer systems in planar, not-necessarily-bipartite, graphs. While perfect matchings on planar graphs can be counted exactly in polynomial time, counting non-perfect matchings was shown by Jerrum (J Stat Phys 1987) to be #P-hard, who also raised the question of whether efficient approximate counting is possible. We answer this affirmatively by showing that the multi-site Glauber dynamics on the set of monomers in a monomer-dimer system always mixes rapidly, and that this dynamics can be implemented efficiently on downward-closed families of graphs where counting perfect matchings is tractable. As further applications of our results, we show how to sample efficiently using multi-site Glauber dynamics from partition-constrained strongly Rayleigh distributions, and nonsymmetric determinantal point processes. In order to analyze mixing properties of the multi-site Glauber dynamics, we establish two notions for generating polynomials of discrete set-valued distributions: sector-stability and fractional log-concavity. These notions generalize well-studied properties like real-stability and log-concavity, but unlike them robustly degrade under useful transformations applied to the distribution. We relate these notions to pairwise correlations in the underlying distribution and the notion of spectral independence introduced by Anari et al. (FOCS 2020), providing a new tool for establishing spectral independence based on geometry of polynomials. As a byproduct of our techniques, we show that polynomials avoiding roots in a sector of the complex plane must satisfy what we call fractional log-concavity; this generalizes a classic result established by Gårding (J Math Mech 1959) who showed homogeneous polynomials that have no roots in a half-plane must be log-concave over the positive orthant.