Randomized approaches to combinatorial packing and covering problems

许多重要的问题可以表述为覆盖和包装问题。这使得从算法和结构的角度研究这类问题至关重要。离散概率论的最新思想和方法在该领域取得了若干突破，有可能重塑该领域。在这里，填充问题的任务是将给定结构中尽可能多的元素组织成合适的不相交的子结构。在覆盖问题中，任务是用尽可能少的合适的子结构（不一定是不相交的）覆盖给定结构的所有元素。这些问题可以被看作是彼此的“双重性”。特别是，它们的解有时重合，在这种情况下，我们得到给定结构的分解。在设计理论中出现了一个经典的例子：在适当的可分性条件下，Steiner三重系统将完全图的边分解为边不相交的三角形，即三角形填充问题的最优解等于三角形覆盖问题的最优解。这些问题有很长的历史，可以追溯到19世纪，并且有许多应用，例如测试，统计设计和编码理论。该项目旨在解决长期存在的关于非完全主图中最优包装的问题。它们在通信网络等方面都有应用。另一个相关的方向是研究涉及更复杂结构的包装和覆盖物，如生成树和可解析设计。我们打算为这样的问题提供一个强大的算法工具，它可以被看作是著名的爆炸引理的近最优打包版本。这将有许多结构和算法应用。此外，包装和覆盖观点在更广泛的上下文中可能是有价值的。特别地，我们提出用它来研究布尔可满足性问题k-SAT。k-SAT问题在计算上很棘手，在理论计算机科学中具有重要的基础意义。再一次，概率的观点被证明是无价的。通过将k-SAT问题表述为一个超图覆盖问题，我们将致力于解决关于随机k-SAT公式统计性质的关键开放问题。

Many important questions can be formulated as covering and packing problems. This makes the study of such problems crucial both from an algorithmic and a more structural point of view. Recent ideas and methods from discrete probability theory have led to several breakthroughs in the area, with the potential to reshape the field. Here in a packing problem the task is to organize as many elements of a given structure as possible into suitable disjoint substructures. In a covering problem the task is to cover all elements of a given structure by as few as possible suitable substructures (which are not necessarily disjoint). These problems can be viewed as "duals" of each other. In particular, their solutions sometimes coincide, in which case we obtain a decomposition of the given structure.A classical example occurs in design theory: under suitable divisibility conditions a Steiner Triple System gives a decomposition of the edges of a complete graph into edge-disjoint triangles, i.e. the optimal solution of the triangle packing problem equals that of the triangle covering problem. Such problems have a long history going back to the 19th century and have many applications, e.g. to testing, statistical design and coding theory. The project intends to approach long-standing questions concerning optimal packings in non-complete host graphs. These have applications e.g. in communication networks. Another related strand is to study packings and coverings involving more complex structures such as spanning trees and resolvable designs. We intend to provide a powerful algorithmic tool for such questions which can be viewed as near-optimal packing version of the famous Blow-up Lemma. This would have numerous structural and algorithmic applications. Furthermore, the packing and covering viewpoint can be valuable in a much broader context. In particular, we propose to employ it to study the Boolean Satisfiability Problem k-SAT. The k-SAT problem is computationally intractable and of fundamental importance in theoretical computer science. Again, the probabilistic perspective has proved to be invaluable. By formulating the k-SAT problem as a hypergraph covering problem, we will aim to solve key open problems regarding statistical properties of random k-SAT formulas.