有界度图上单二聚系统的次线性时间算法
Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs
摘要：
对于图$G$，令$Z(G,λ)$是由$\sum_k m_k(G)λ^k$定义的单体-二聚体系统的配分函数，其中$m_k(G)$是$G$中大小为$k$的匹配数。我们考虑有界度的图，并开发了一个次线性时间算法估计$\log Z(G,λ)$在任意值$λ>0$内添加剂误差$εn$具有高概率。该算法的查询复杂度不依赖于$G$的大小，在$1/ε$中是多项式的，并给出了该问题在$1/ε$中的二次下界.这是第一次分析一个$# P$-完全问题的次线性时间近似算法。我们的方法是基于与$Z(G,λ)$相关的吉布斯分布的相关衰减。我们表明，我们的算法近似的概率顶点被覆盖的匹配，根据吉布斯分布采样，在一个接近最佳的次线性时间。我们扩展我们的结果，以近似的平均大小和熵的这样一个匹配的添加剂内的错误与高概率，再次查询的复杂性是多项式在$1/ε$和下限是二次在$1/ε$。我们的算法实现简单，在处理海量数据集时具有实用价值。我们的研究结果扩展到其他系统的相关性衰减是已知的独立设置问题的关键活动举行。
英文摘要：
For a graph $G$, let $Z(G,λ)$ be the partition function of the monomer-dimer system defined by $\sum_k m_k(G)λ^k$, where $m_k(G)$ is the number of matchings of size $k$ in $G$. We consider graphs of bounded degree and develop a sublinear-time algorithm for estimating $\log Z(G,λ)$ at an arbitrary value $λ>0$ within additive error $εn$ with high probability. The query complexity of our algorithm does not depend on the size of $G$ and is polynomial in $1/ε$, and we also provide a lower bound quadratic in $1/ε$ for this problem. This is the first analysis of a sublinear-time approximation algorithm for a $# P$-complete problem. Our approach is based on the correlation decay of the Gibbs distribution associated with $Z(G,λ)$. We show that our algorithm approximates the probability for a vertex to be covered by a matching, sampled according to this Gibbs distribution, in a near-optimal sublinear time. We extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability, where again the query complexity is polynomial in $1/ε$ and the lower bound is quadratic in $1/ε$. Our algorithms are simple to implement and of practical use when dealing with massive datasets. Our results extend to other systems where the correlation decay is known to hold as for the independent set problem up to the critical activity.
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