Combinational, Structural and algorithmic aspects of temporal graphs

时间图的组合、结构和算法方面

• 批准号: 2903280
• 金额: --
• 依托单位: University of Liverpool
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2903280
• 关键词: Combinational; Structural; algorithmic; aspects; temporal

Boolean matrices (i.e., 0/1-matrices) appear in different contexts in computer science, machine learning, mathematics, and other areas. For example, in graph theory they provide a standard way to represent graphs, via so called adjacency and incidence matrices. In communication complexity, the main objects of study, communication problems, are described by Boolean matrices [9, 11]. In learning theory, Boolean matrices are used to represent hypothesis classes [10]. In all these areas, researchers used Boolean combinations of Boolean matrices as a natural way to express new objects via "simpler" objects. Usually, the goal is to conclude that if some property holds for the "simpler" objects, then it can or cannot be extended to hold for Boolean combinations of these objects. In graph theory, this approach was recently used in the context of graph labelling schemes [4, 7]. In communication complexity, it is used to study communication protocols with access to oracles [6, 7, 8]. In learning theory, it is used to combine hypothesis classes to obtain more powerful ones [2]. Thus, Boolean combinations of Boolean matrices naturally appear as a useful tool in a number of areas. However, (1) their usage is ad-hoc; and (2) their studies in different areas are mostly independent. The central aim of the present project is to: 1. Systematically study Boolean combinations in the context of graph theory as a means of expressing graph classes via some other graph classes. The main intention of this is to understand which graph properties are preserved by Boolean combinations, and under what conditions.The secondary aims of the project are to: 2. Formulate results about Boolean combinations of Boolean matrices known in different areas using a common language and where appropriate transfer such results from one area to another; this will contribute to building links between different areas of computer science and mathematics.3. Initiate a systematic study of Boolean combinations of Boolean matrices independently of their application.We will investigate which Boolean functions on certain hereditary classes preserve particular graph properties. Cases of these have been studied before, for example the boxicity of a graph, G, is the minimum number of interval graphs needed to represent G as their intersection [12], the number of complete graphs needed to represent graphs as their XOR [3], the number of complete or equivalence graphs needed to represent graphs as their union [1] and the number of threshold graphs needed to represent n-vertex graphs as certain functions of them [5]. We will be looking at graph functions more generally.References[1] Noga Alon. Covering graphs by the minimum number of equivalence relations. Combinatorica, 6(3):201-206, 1986. [2] Noga Alon, Amos Beimel, Shay Moran, and Uri Stemmer. Closure properties for private classification and online prediction. In Conference on Learning Theory, pages 119-152. PMLR, 2020. [3] Calum Buchanan, Christopher Purcell, and Puck Rombach. Subgraph complementation and minimum rank. The Electronic Journal of Combinatorics, 29(1), 2022. [4] Maurice Chandoo. Logical labeling schemes. Discrete Mathematics, 346(10):113565, 2023. [5] Paul Erdös, Edward T Ordman, and Yechezkel Zalcstein. Bounds on threshold dimension and disjoint threshold coverings. SIAM Journal on Algebraic Discrete Methods, 8(2):151-154, 1987. [6] Yuting Fang, Lianna Hambardzumyan, Nathaniel Harms, and Pooya Hatami. No complete problem for constant-cost randomized communication. In Proceedings of the Symposium on Theory of Computing (STOC 2024), 2024. [7] Nathaniel Harms, Sebastian Wild, and Viktor Zamaraev. Randomized communication and implicit graph representations. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022), pages 1220-1233, 2022. [8] Nathaniel Harms and Viktor Zamaraev. Randomized communication and implicit representations for matrices and graphs of sm

布尔材料（即0/1-Matrices）出现在计算机科学，机器学习，数学和其他领域的不同情况下。例如，在图理论中，它们通过所谓的邻接和发射率提供了一种标准方式来表示图形。在沟通复杂性中，研究的主要对象，沟通问题是由布尔物质描述的[9，11]。在学习理论中，布尔物质用于代表假设类别[10]。在所有这些领域，研究人员都将布尔材料的布尔组合用作通过“简单”对象表达新对象的自然方式。通常，目标是包括，如果某些属性具有“简单”对象，则可以或不能将其扩展以使其对这些对象的布尔组合保持。在图理论中，最近在图标记方案的背景下使用了这种方法[4，7]。在通信复杂性中，它用于研究访问甲壳的通信协议[6,7,8]。在学习理论中，它用于结合假设类以获得更强大的类别[2]。这是布尔材料的布尔组合自然是在许多领域的有用工具。但是，（1）它们的使用是临时的； （2）他们在不同领域的研究主要是独立的。本项目的核心目的是：1。系统地研究图理论的背景下的布尔组合，以此作为通过其他一些图类别表达图形类的手段。这样做的主要目的是了解哪些图形属性是由布尔组合保留的，在哪个条件下。这将有助于在计算机科学和数学的不同领域之间建立联系3。启动对布尔物质的布尔材料组合的系统研究，独立于其应用。我们将调查哪些布尔在某些遗传类别上的功能保留特定的图形特性。以前已经对这些情况进行了研究，例如图形G的拳头性是表示G表示为g所需的最小间隔图数量[12]，将图表示为XOR [3]所需的完整图数，所需的完整或等价图的数量表示图表表示图形[1]以及表示截止图的数量N-vers n-vers n-vers n n sefress n-vers n-vers n-vers n-vers n-vers n sefress n-verss n sefress n-verss as thement and verts n s y-vers ns-verscess n-verscess n and-versces as表示。我们将更广泛地研究图形函数[1] Noga Alon。按照最小数量的等效关系覆盖图。 Combinatorica，6（3）：201-206，1986。[2] Noga Alon，Amos Beimel，Shay Moran和Uri Stemmer。关闭私人分类和在线预测的关闭属性。在学习理论会议上，第119-152页。 PMLR，2020。[3] Calum Buchanan，Christopher Purcell和Puck Rombach。子图汇编和最低等级。 Electronic of Compinatorics，29（1），2022。[4] Maurice Chandoo。逻辑标签方案。离散数学，346（10）：113565，2023。[5]Paulerdös，Edward T Ordman和Yechezkel Zalcstein。阈值尺寸和脱节阈值覆盖范围的边界。 SIAM关于代数离散方法的杂志，8（2）：151-154，1987。[6] Yuting Fang，Lianna Hambardzumyan，Nathaniel Harms和Pooya Hatami。恒定成本随机通信没有完整的问题。在计算理论研讨会论文集（Stoc 2024），2024。[7] Nathaniel Harms，Sebastian Wild和Viktor Zamaraev。随机通信和隐式图表。在第54届年度ACM SIGACT计算理论研讨会上（Stoc 2022），第1220-1233、2022页。[8] Nathaniel Harms和Viktor Zamaraev。 SM的物品和图形的随机通信和隐式表示