An emerging trend in approximate counting is to show that certain `low-temperature' problems are easy on typical instances, despite worst-case hardness results. For the class of regular graphs one usually shows that expansion can be exploited algorithmically, and since random regular graphs are good expanders with high probability the problem is typically tractable. Inspired by approaches used in subexponential-time algorithms for Unique Games, we develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition. In such graphs it may not suffice to explore the state space of the model close to ground states, and a novel feature of our method is to efficiently find a larger set of `pseudo-ground states' such that it is enough to explore the model around each pseudo-ground state.

翻译：近似计数领域的一个新兴趋势是表明，尽管存在最坏情况下的复杂度难题，但某些“低温”问题在典型实例上其实是容易处理的。对于正则图类，通常可以证明扩张性能够被算法利用，而由于随机正则图以高概率为优质扩展图，因此该问题通常是易处理的。受用于唯一游戏（Unique Games）的次指数时间算法方法的启发，我们针对满足小集扩展条件的图上的铁磁Potts模型的配分函数，开发了一种近似算法。在这类图中，仅探索模型在基态附近的状态空间可能不够，我们方法的一个新颖之处在于高效找出一个更大的“伪基态”（pseudo-ground states）集合，从而只需在每个伪基态附近探索模型即可。