Recently, Rowe and Aishwaryaprajna (2019) introduced a simple majority vote technique that efficiently solves Jump with large gaps, OneMax with large noise, and any monotone function with a polynomial-size image. In this paper, we identify a pathological condition for this algorithm: the presence of spin-flip symmetry in the problem instance. Spin-flip symmetry is the invariance of a pseudo-Boolean function to complementation. Many important combinatorial optimization problems admit objective functions that exhibit this pathology, such as graph problems, Ising models, and variants of propositional satisfiability. We prove that no population size exists that allows the majority vote technique to solve spin-flip symmetric functions of unitation with reasonable probability. To remedy this, we introduce a symmetry-breaking technique that allows the majority vote algorithm to overcome this issue for many landscapes. This technique requires only a minor modification to the original majority vote algorithm to force it to sample strings in {0,1}n${0,1}n$${0,1}n$ from a dimension n−1$n-1$$n-1$ hyperplane. We prove a sufficient condition for a spin-flip symmetric function to possess in order for the symmetry-breaking voting algorithm to succeed, and prove its efficiency on generalized TwoMax, a spin-flip symmetric variant of Jump, and families of constructed 3-NAE-SAT and 2-XOR-SAT formulas. We also prove that the algorithm fails on the one-dimensional Ising model, and suggest different techniques for overcoming this. Finally, we present empirical results that explore the tightness of the runtime bounds and the performance of the technique on randomized satisfiability variants.

最近，Rowe 和 Aishwaryaprajna (2019) 引入了一种简单多数投票技术，可以有效解决具有大间隙的Jump 、具有大噪声的OneMax以及具有多项式大小图像的任何单调函数。在本文中，我们确定了该算法的病态条件：问题实例中存在自旋翻转对称性。自旋翻转对称性是伪布尔函数对补集的不变性。许多重要的组合优化问题都承认表现出这种病态的目标函数，例如图问题、伊辛模型和命题可满足性的变体。我们证明不存在允许多数投票技术以合理概率求解统一的自旋翻转对称函数的总体规模。为了解决这个问题，我们引入了一种对称性破坏技术，该技术允许多数投票算法在许多景观中克服这个问题。该技术只需要对原始多数投票算法进行较小的修改即可强制其对字符串进行采样{ 0 , 1 }n${0,1}n$${0,1}n$从一个维度n − 1$n-1$$n-1$超平面。我们证明了自旋翻转对称函数满足对称性破缺投票算法成功的充分条件，并证明了其在广义 TwoMax（Jump 的自旋翻转对称变体）和构造的 3-NAE系列上的效率-SAT 和 2-XOR-SAT 公式。我们还证明该算法在一维 Ising 模型上失败，并提出了克服此问题的不同技术。最后，我们提出了实证结果，探索运行时界限的严格性以及该技术在随机可满足性变体上的性能。