Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint to design contention resolution schemes, which avoids to deal explicitly with the randomization in the second step. This is achieved by focusing on the marginals of a dropping procedure. Apart from avoiding one source of randomization, our viewpoint allows for employing polyhedral techniques. Both can significantly simplify the construction and analysis of contention resolution schemes. We show how, through our framework, one can obtain an optimal monotone contention resolution scheme for bipartite matchings. So far, only very few results are known about optimality of monotone contention resolution schemes. Our contention resolution scheme for the bipartite case also improves the lower bound on the correlation gap for bipartite matchings. Furthermore, we derive a monotone contention resolution scheme for matchings that significantly improves over the previously best one. At the same time, our scheme implies that the currently best lower bound on the correlation gap for matchings is not tight. Our results lead to improved approximation factors for various constrained submodular function maximization problems over a combination of matching constraints with further constraints.

中文翻译：

---

一种基于多面体视点的二分匹配最优单调竞争解决方案

松弛和舍入方法成为约束子模函数最大化的标准和极其通用的工具。在这种情况下，最常见的舍入技术之一是争用解决方案。此类方案通过首先独立地舍入每个坐标，然后丢弃一些元素以达到可行集来舍入小数点。此外，删除元素的第二步通常是随机的。这导致程序中的随机化的额外来源，这可能使分析复杂化。我们建议采用不同的多面体观点来设计争用解决方案，从而避免在第二步中明确处理随机化。这是通过关注丢弃过程的边缘来实现的。除了避免一种随机化来源之外，我们的观点允许采用多面体技术。两者都可以显着简化争用解决方案的构建和分析。我们展示了如何通过我们的框架获得用于二分匹配的最佳单调竞争解决方案。到目前为止，关于单调竞争解决方案的最优性的结果知之甚少。我们针对二分情况的争用解决方案还改进了二分匹配的相关差距的下限。此外，我们推导出了一种单调的竞争解决方案，该方案显着改善了以前最好的匹配方案。同时，我们的方案意味着匹配的相关差距的当前最佳下限并不严格。