Many problems in applied computer science can be expressed in a graph setting and solved by finding an appropriate vertex labeling of the associated graph. It is also common to identify the term “appropriate labeling” with a labeling that optimizes some application-motivated objective function. The goal of this work is to present two algorithms that, for the objective functions in a general format motivated by image processing tasks, find such optimal labelings. Specifically, we consider a problem of finding an optimal binary labeling for the objective function defined as the max-norm over a set of local costs of a form that naturally appears in image processing. It is well known that for a limited subclass of such problems, globally optimal solutions can be found via watershed cuts, that is, by the cuts associated with the optimal spanning forests of a graph. Here, we propose two new algorithms for optimizing a broader class of such problems. The first algorithm, that works for all considered objective functions, returns a globally optimal labeling in quadratic time with respect to the size of the graph (i.e., the number of its vertices and edges) or, for an image associated graph, the size of the image. The second algorithm is more efficient, with quasi-linear time complexity, and returns a globally optimal labeling provided that the objective function satisfies certain given conditions. These conditions are analogous to the submodularity conditions encountered in max-flow/min-cut optimization, where the objective function is defined as sum of all local costs. We will also consider a refinement of the max-norm measure, defined in terms of the lexicographical order, and examine the algorithms that could find minimal labelings with respect to this refined measure.

中文翻译：

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优化基于最大范数的目标函数的两种多项式时间图标记算法

应用计算机科学中的许多问题都可以在图形设置中表达，并可以通过找到相关图形的适当顶点标记来解决。用优化某些应用程序驱动的目标功能的标签来标识“适当的标签”也是很常见的。这项工作的目标是提出两种算法，对于以图像处理任务为动力的通用格式的目标函数，可以找到这种最佳标记。具体来说，我们考虑一个问题，即为目标函数找到最佳二进制标记，该目标函数定义为在图像处理中自然出现的表单的一组局部成本上的最大范数。众所周知，对于此类问题的有限子类，可以通过分水岭切割找到全球最佳解决方案，即通过与图的最佳生成林相关联的切割。在这里，我们提出了两种新算法来优化这类问题的范围。适用于所有考虑的目标函数的第一种算法，相对于图的大小（即，其顶点和边的数量）或图像关联图的大小，以二次时间返回全局最优标记。图片。第二种算法效率更高，具有准线性时间复杂度，并且在目标函数满足某些给定条件的情况下返回全局最优标记。这些条件类似于最大流量/最小切割优化中遇到的次模量条件，其中目标函数定义为所有本地成本的总和。我们还将考虑对max-norm量度进行改进，