The minimum graph cut and minimum $s$-$t$-cut problems are important primitives in the modeling of combinatorial problems in computer science, including in computer vision and machine learning. Some of the most efficient algorithms for finding global minimum cuts are randomized algorithms based on Karger's groundbreaking contraction algorithm. Here, we study whether Karger's algorithm can be successfully generalized to other cut problems. We first prove that a wide class of natural generalizations of Karger's algorithm cannot efficiently solve the $s$-$t$-mincut or the normalized cut problem to optimality. However, we then present a simple new algorithm for seeded segmentation / graph-based semi-supervised learning that is closely based on Karger's original algorithm, showing that for these problems, extensions of Karger's algorithm can be useful. The new algorithm has linear asymptotic runtime and yields a potential that can be interpreted as the posterior probability of a sample belonging to a given seed / class. We clarify its relation to the random walker algorithm / harmonic energy minimization in terms of distributions over spanning forests. On classical problems from seeded image segmentation and graph-based semi-supervised learning on image data, the method performs at least as well as the random walker / harmonic energy minimization / Gaussian processes.
翻译:最起码的图形剪切和最低的美元-美元-美元-美元-美元-美元-美元-美元-问题是计算机科学组合问题模型中的重要原始因素,包括计算机视觉和机器学习。找到全球最低削减率的最有效算法是基于Karger的开创性收缩算法的随机化算法。在这里,我们研究Karger的算法能否成功地推广到其他切割问题。我们首先证明,卡杰的计算法的广泛自然概括性无法有效地解决美元-美元-美元-美元-美元-切割或标准化的切除问题至最佳性。然而,我们随后为种子化分解/基于图形的半监督性学习提供了一种简单的新算法,这种算法以Karger的原始算法为基础,显示对这些问题,Karger的算法的延伸可能有用。新的算法具有线性无线性、随机性运行时间,并产生一种潜力,可以被解释为属于某个种子/类的样本的远端概率。我们澄清了它与随机演算法/协调能源-和谐度-最小化能源最小化的节能-最小化精度-最小化分法分布于模型/最难的图像/结构,在正正态/结构图像图上,在透式图像上,在原始结构-最慢的分布图/最慢的流化的图像中进行。