六顶点模型的复杂性分类

六顶点模型的可逼近性

Approximability of the Six-vertex Model

在本文中，我们采取的第一步对六顶点模型，在统计物理学中广泛研究的对象的近似复杂性的分类。我们的复杂性结果符合物理学中的相变现象。我们发现，六顶点模型的近似复杂性表现出显着不同的两侧分离的相变阈值。此外，我们提出的结构特性的六顶点模型的平面图的参数设置，有已知的关系的Tutte多项式$T(G, x, y)$。

Beyond Windability: An FPRAS for The Six-Vertex Model

The six-vertex model is an important model in statistical physics and has deep connections with counting problems. There have been some fully polynomial randomized approximation schemes (FPRAS) for the six-vertex model [30, 10], which all require that the constraint functions are windable. In the present paper, we give an FPRAS for the six-vertex model with an unwindable constraint function by Markov Chain Monte Carlo method (MCMC). Different from [10], we use the Glauber dynamics to design the Markov Chain depending on a circuit decomposition of the underlying graph. Moreover, we prove the ra…

New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification

We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models on planar graphs, assuming #P is not P. This leads to the following exact complexity classification: For every parameter setting in ${\mathbb C}$ for the six-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) #P-hard for general graphs but computable in P-time for planar graphs, or (3) #P-hard even for planar gra…

边诱导顶点割的复杂性及其应用

Complexity and Applications of Edge-Induced Vertex-Cuts

受超图分解算法的启发，我们引入了边诱导顶点割的概念，并将其与著名的边割和顶点割的概念进行了比较。我们研究了计算最小边诱导顶点割的复杂性，并通过在网络可靠性和约束满足方面的应用证明了我们的概念的实用性。