Structure in Designs, Coverings and Decompositions

设计、覆盖和分解的结构

• 批准号: RGPIN-2022-03816
• 负责人: Danziger, Peter
• 金额: $ 1.53万
• 依托单位: Ryerson University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758501
• 关键词: Structure; Designs; Coverings; Decompositions

This application is for the renewal of my discovery grant which will facilitate ongoing research activities. It will serve to support HQP, promote collaborative projects and enhance the dissemination of related results. Combinatorial designs and related graph decompositions and factorizations, as well as packings and coverings, 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 and related objects and 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. I have long been involved with finding various kinds of factorizations, initially uniformly 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 the related area of Covering Arrays. This project will support my work in all of these areas. The proposal will use state of the art combinatorial techniques as well advanced algorithmic methods to investigate these structures. It will also provide ample opportunity for HQP training in these areas an beyond. The longstanding Oberwolfach problem, introduced by Ringel in the 1960s has received much attention over the years, with several recent advances. The related Hamilton Waterloo problem, where I have had significant success, requires a factorization of the complete graph into a variety of cycle types. One of the objectives of this project is to build on my recent results and continue investigating these and other related problems. Another goal is to undertake a consideration of combinatorial objects which fail to have a particular structure, such as resolvability, or have excess structure, such as Doubly Resolvable Designs 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 and further generalisations of them. This includes cases with restricted interaction sets and generalisations to Sequence Covering Arrays.

这份申请是为了续期我的研究资助，以促进我正在进行的研究活动。它将支持卫生质量规划，促进合作项目，并加强相关成果的传播。组合设计和相关的图分解和因子分解，以及包装和覆盖，为理解复杂离散结构（如网络）的相互作用特性提供了一种理想的方法，也是研究此类结构的“互联性”特性的一种方法。提出的研究计划的中心焦点是对组合设计和相关对象的结构的调查，并考虑具有特定结构或缺乏结构的设计。该领域的发展将提供对所涉及对象的结构的更深层次的理解，以及对其他组合问题的见解。设计在统计学、编码理论和调度方面有着众所周知的应用。此外，在算法效率、软件和网络测试等多种问题上也有潜在的应用。长期以来，我一直致力于寻找各种分解，最初是统一的和类一致的可分解设计，将这项工作扩展到更一般的分解，最著名的是循环分解。我也考虑过因式分解不可能的情况。最近，我对覆盖阵列的相关领域产生了兴趣。这个项目将支持我在所有这些领域的工作。该提案将使用最先进的组合技术以及先进的算法方法来研究这些结构。它还将为这些领域及其他领域的HQP培训提供充足的机会。Ringel在20世纪60年代提出的长期存在的Oberwolfach问题多年来受到了广泛关注，最近取得了一些进展。在相关的汉密尔顿滑铁卢问题中，我取得了显著的成功，它需要将完全图分解成各种循环类型。这个项目的目标之一是建立在我最近的结果，并继续调查这些和其他相关的问题。另一个目标是考虑没有特定结构的组合对象，比如可解析性，或者有多余的结构，比如双重可解析设计，覆盖数组由于在测试中的应用，尤其是软件和网络测试中，最近受到了很多关注。本提案的目标之一是进一步研究这些对象并进一步概括它们。这包括具有受限交互集和泛化到序列覆盖数组的情况。