This paper concerns the efficient implementation of a method for optimal binary labeling of graph vertices, originally proposed by Malmberg and Ciesielski (2020). This method finds, in quadratic time with respect to graph size, a labeling that globally minimizes an objective function based on the $L_\infty$-norm. The method enables global optimization for a novel class of optimization problems, with high relevance in application areas such as image processing and computer vision. In the original formulation, the Malmberg-Ciesielski algorithm is unfortunately very computationally expensive, limiting its utility in practical applications. Here, we present a modified version of the algorithm that exploits redundancies in the original method to reduce computation time. While our proposed method has the same theoretical asymptotic time complexity, we demonstrate that is substantially more efficient in practice. Even for small problems, we observe a speedup of 4-5 orders of magnitude. This reduction in computation time makes the Malmberg-Ciesielski method a viable option for many practical applications.

翻译：本文关注Malmberg和Ciesielski（2020）提出的图顶点最优二值标记方法的高效实现。该方法以图大小的二次时间找到全局最小化基于$L_\infty$-范数的目标函数的标记，从而实现对一类具有高相关性的优化问题（如图像处理和计算机视觉应用领域）的全局优化。在原始公式中，Malmberg-Ciesielski算法计算代价极高，限制了其在实际应用中的实用性。本文提出该算法的改进版本，通过利用原始方法的冗余性减少计算时间。尽管所提方法具有相同的理论渐近时间复杂度，但实验证明其在实际应用中效率显著提升。即使对于小规模问题，我们也观察到4-5个数量级的加速。这种计算时间的缩减使Malmberg-Ciesielski方法成为众多实际应用的可行选择。