Structure in designs, coverings and decompositions

组合设计为理解复杂离散结构(如网络)的相互作用属性提供了一种理想的方法，也是研究这种结构的“互连性”属性的一种方法。拟议的研究计划的中心焦点是调查组合设计的结构；考虑具有特定结构的设计，或不具有特定结构的设计。这一领域的发展将提供对所涉及对象的结构的更深入的理解，以及对其他组合问题的洞察。设计在统计学、编码理论和调度方面有着广泛的应用，此外，在算法效率、软件和网络测试以及数值分析等问题上也有潜在的应用。我长期以来一直致力于寻找各种分解，初始一致和类一致可分解的设计，将这项工作扩大到包括更一般的分解，最著名的是循环分解。我还考虑过不可能进行因式分解的情况。最近，我开始对介绍数组感兴趣。这个项目将支持我在所有这些领域的工作。 Oberwolfach问题由来已久，由Ringel在20世纪60年代作为座椅问题提出，多年来受到了极大的关注，最近取得了一些进展。更一般的哈密尔顿-滑铁卢问题需要将完全图因式分解成各种圈类型。这个项目的目标之一就是继续调查这些问题。另一个目标是考虑不具有特定结构的组合对象，例如可分解性。 由于覆盖阵列在测试中的应用，特别是在软件和网络测试中的应用，最近引起了人们的极大兴趣。这项提议的目标之一是进一步调查这些物体，特别是关于对它们可能展示的测试有用的子结构。这包括混合强度覆盖阵列和具有禁止对或其他配置的覆盖阵列，以及与合作者一起开发用于生成覆盖阵列的在线资源，这是一种重要的工业资源。

Combinatorial Designs provide an ideal way to understand the interaction properties of complex discrete structures, such as networks; a way to investigate the `interconnectedness' properties of such structures. The central focus of the proposed research program is the investigation of the structure of combinatorial designs; to consider designs with a particular structure, or lack thereof. Development in this area will provide a deeper understanding of the structure of the objects involved as well as insight into other combinatorial questions. Designs have well-known applications to statistics, coding theory and scheduling, in addition, there are potential applications to such diverse questions as algorithmic efficiency, software and network testing and numerical analysis. I have long been involved with finding various kinds of factorizations, initially uniform and class-uniformly resolvable designs, broadening this work to include more general factorizations, most notably cycle factorizations. I have also considered cases where factorization is not possible. More recently, I have become interested in Covering Arrays. This project will support my work in all of these areas. The longstanding Oberwolfach problem, introduced by Ringel in the 1960s as a seating problem has recieved much attention over the years, with several recent advances. The more general Hamilton-Waterloo problem requires a factorization of the complete graph into a variety of cycle types. One of the objectives of this project is to continue investigation into these problems. Another goal is to undertake a consideration of combinatorial objects which fail to have a particular structure, such as resolvability. Covering arrays have received much interest of late due to their applications in testing, particularly software and network testing. One of the goals of this proposal is to further investigate these objects, particularly with respect to substructures useful for testing that they may exhibit. This includes, mixed strength covering arrays and covering arrays with forbidden pairs or other configurations as well as the development, together with collaborators, of an online resource for generation of covering arrays, an important industrial resource.