Problems in Randomized Algorithms, Random Graphs, and Computational Geometry

随机算法、随机图和计算几何中的问题

• 批准号: RGPIN-2019-04269
• 负责人: Delcourt, Michelle
• 金额: $ 2.4万
• 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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737521
• 关键词: Problems; Randomized; Algorithms; Random; Graphs

The overarching goal of my research program is to solve a number of long standing open problems in the intersection of theoretical computer science, graph theory, and probability theory. A commonality between the proposed topics is the use of probabilistic techniques. More precisely, my proposed research program focuses on the following areas. (Objective 1) Randomized algorithms. Although counting the number of proper k-colourings of a graph is a computationally hard problem, Jerrum, Valiant, and Vazirani showed that a nearly uniform sampler gives rise to an approximate enumeration, motivating the question of finding an algorithm to efficiently generate uniformly random proper colourings of a graph; this is a central topic in both computer science and statistical physics.  My colleagues and I made a recent breakthrough, the first progress on the most important question in this area in 19 years. My students and I will push this approach further, both working on the fundamental problem for Glauber dynamics and using similar techniques to bound the mixing time of other Markov chains as well. (Objective 2) Random regular graphs. A question that has attracted much interest in graph theory is: under what conditions can we partition the edge set of a graph into edge disjoint copies of a subgraph? This is fundamentally related to some of the most notorious open areas of research such as finding orientations of certain types, nowhere-zero flows, and colourings of planar graphs. A key new insight is that moving long standing problems from structural graph theory to the random regular setting can provide additional machinery and help to shed light on classical, longstanding problems.  For instance using probabilistic techniques, a coauthor and I recently showed that a random 4-regular graph has a decomposition into 3-stars asymptotically almost surely. An important line of research with far reaching applications is generalizing this result in various ways.  My students and I will pursue developing methods for k-stars in d-regular random graphs as well as decompositions into other trees. (Objective 3) Computational geometry. An active line of inquiry in combinatorics in recent years has been extending classical results to the so-called sparse random setting, where the goal is to show that certain known properties of "dense" combinatorial structures are inherited by their randomly chosen "sparse" substructures. In this spirit my team and I will develop an algorithmic approach that shows if a given algebraic hypergraph is "dense" in a certain sense, then a generic low-dimensional subset of the vertices induces a subhypergraph that is also "dense." Such results have applications in computational geometry and matroid theory. My team and I will also establish a natural generalization of the classical dimension of fibers theorem in algebraic geometry, a result interesting in its own right.

我的研究计划的总体目标是解决理论计算机科学、图论和概率论交叉领域中一些长期存在的开放性问题。所提议主题之间的共同点是概率技术的使用。 更准确地说，我提出的研究计划重点关注以下领域。 （目标 1）随机算法。尽管计算图的正确 k 着色的数量是一个计算难题，但 Jerrum、Valiant 和 Vazirani 表明，几乎均匀的采样器会产生近似枚举，从而激发了寻找一种算法来有效生成图的均匀随机正确着色的问题；这是计算机科学和统计物理学的中心主题。  我和我的同事最近取得了突破，这是19年来在这一领域最重要问题上的首次进展。我和我的学生将进一步推动这种方法，既致力于解决格劳伯动力学的基本问题，又使用类似的技术来限制其他马尔可夫链的混合时间。 （目标 2）随机正则图。 图论中引起人们极大兴趣的一个问题是：在什么条件下我们可以将图的边集划分为子图的边不相交副本？这从根本上与一些最臭名昭著的开放研究领域相关，例如寻找某些类型的方向、无处零流和平面图的着色。一个重要的新见解是，将长期存在的问题从结构图论转移到随机规则设置可以提供额外的机制，并有助于阐明经典的、长期存在的问题。  例如，我和一位合著者最近使用概率技术证明，随机 4 正则图几乎肯定会渐近分解为 3 星。具有深远应用的一个重要研究方向是以各种方式推广这一结果。  我和我的学生将致力于开发 d 正则随机图中的 k 星以及分解为其他树的方法。 （目标 3）计算几何。近年来，组合学领域的一个活跃的研究方向一直是将经典结果扩展到所谓的稀疏随机设置，其目标是证明“密集”组合结构的某些已知属性是由其随机选择的“稀疏”子结构继承的。本着这种精神，我和我的团队将开发一种算法方法，该方法表明如果给定的代数超图在某种意义上是“稠密”的，那么顶点的通用低维子集会产生一个也是“稠密”的子超图。这些结果在计算几何和拟阵理论中都有应用。我和我的团队还将在代数几何中建立纤维定理的经典维数的自然推广，这一结果本身就很有趣。