There are many optimization problems having the following common property: Given a total task consisting of many subtasks, the problem asks to find a solution to complete only part of these subtasks. Examples include the k-Forest problem and the k-Multicut problem, etc. These problems are called partial optimization problems, which are often NP-hard. In this paper, we systematically study the LP-rounding plus greed approach, a method to design approximation algorithms for partial optimization problems. The approach is simple, powerful and versatile. We show how to use this approach to design approximation algorithms for the k-Forest problem, the k-Multicut problem, the k-Generalized connectivity problem, etc.

有许多优化问题具有以下共同特性：给定一个由许多子任务组成的总任务，该问题要求找到仅完成这些子任务中的一部分的解决方案。例子包括k- Forest 问题和k- Multicut 问题等。这些问题被称为部分优化问题，通常是NP-hard 问题。在本文中，我们系统地研究了 LP 舍入加贪婪方法，这是一种设计用于部分优化问题的近似算法的方法。该方法简单、强大且用途广泛。我们展示了如何使用这种方法为k- Forest 问题、k- Multicut 问题、k-广义连通性问题等。