In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set $ S $, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set $ Q \subseteq \mathbb{R}^n $ containing a set $ S \subseteq \{0,1\}^n $ by exploiting certain additional information about $ S $. Namely, the required extra information will be in the form of a Boolean formula $ \phi $ defining the target set $ S $. The aim of this work is to analyze various aspects regarding the strength of our procedure. As one result, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.

中文翻译：

---

使用布尔公式加强 0/1 集的凸松弛

在凸整数规划中，已经开发了各种程序来加强整数点集的凸松弛。一方面，有几种通用方法可以在没有集合 $ S $ 的特定知识的情况下加强松弛，例如流行的线性规划或半定规划层次结构。另一方面，已经设计了各种方法来为组合优化中出现的非常特定的集合获得增强的松弛。我们提出了一种新的有效方法，可以在这两种方法之间进行插值。我们的程序通过利用有关 $ S $ 的某些附加信息来增强包含集合 $ S \subseteq \{0,1\}^n $ 的任何凸集 $ Q \subseteq \mathbb{R}^n $。即，所需的额外信息将采用定义目标集 $ S $ 的布尔公式 $\phi $ 的形式。这项工作的目的是分析有关我们程序强度的各个方面。作为一个结果，将我们程序的迭代应用程序解释为层次结构，我们的发现简化、改进和扩展了 Bienstock 和 Zuckerberg 之前在覆盖问题上的结果。