We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. In a recent work, we established NP-hardness of approximating the average size of an independent set utilizing hardness of the corresponding optimization problem and the related phase transition behavior. Here, we instead consider settings where the underlying optimization problem is easily solvable. Our main contribution is to classify the complexity of approximating a wide class of observables via a generic reduction from approximate counting to the problem of estimating local observables. The key idea is to use the observables to interpolate the counting problem. Using this new approach, we are able to study observables on bipartite graphs where the underlying optimization problem is easy but the counting problem is believed to be hard. The most-well studied class of graphs that was excluded from previous hardness results were bipartite graphs. We establish hardness for estimating the average size of the independent set in bipartite graphs of maximum degree 6; more generally, we show tight hardness results for general vertex-edge observables for antiferromagnetic 2-spin systems on bipartite graphs. Our techniques go beyond 2-spin systems, and for the ferromagnetic Potts model we establish hardness of approximating the number of monochromatic edges in the same region as known hardness of approximate counting results.
翻译:我们研究的是估算Gibbs分布的本地可见的计算复杂性。 一个简单的组合示例是图中独立集的平均大小。 在最近的一项工作中, 我们利用相应的优化问题和相关阶段过渡行为的硬性, 确定了独立集的平均大小, 使用相应的优化问题和相关阶段过渡行为的硬性, 以近于独立集的平均大小的NP- 硬性。 这里, 我们考虑的是潜在的优化问题很容易溶解的设置。 我们的主要贡献是通过从粗略点数到估计本地观测的难题来分类一个大类观测的复杂性。 关键的想法是使用可观测到的硬性来对计算问题进行内推。 使用这一新的方法, 我们能够研究双部分图中观察到的硬性结果, 其中潜在的优化问题很简单, 但计数问题被认为是很困难的。 被先前硬性结果排除的最精密的图表类别是两部分的图形。 我们为估算最大度6的双面图中独立组的平均大小而建立硬性模型; 更一般地, 我们为普通的双向系统显示硬性硬性结果, 我们的双向双向的双向系统建立我们双向的双向的双向系统。