Probabilistic and Extremal Combinatorics

概率和极值组合学

• 批准号: 1001638
• 负责人: Tom Bohman
• 金额: $ 27万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001638&HistoricalAwards=false
• 关键词: Probabilistic; Extremal; Combinatorics

The majority of this research is in the area of probabilistic combinatorics, which is comprised of the study of random combinatorial structures, randomized algorithms and probabilistic existence proofs for combinatorics (i.e. the probabilistic method). In recent years, this field has seen the development of a technique for proving that key statistics of a discrete stochastic process are likely to remain close to their expected trajectories as the process evolves. This method is known as the differential equations method for greedy algorithms and random graph processes. The investigator recently extended this technique to analyze the triangle-free process, a `controlled' random graph process that produces an interesting distribution on the space of triangle-free graphs. The investigator showed that a graph drawn from this distribution is likely to have independence number within a constant multiplicative factor of the smallest possible. In other words, the triangle-free process produces a Ramsey R(3,t) graph. The investigator shall continue the development of this method for the study of related `controlled' random processes, with an emphasis on processes that are interesting from the perspective of extremal combinatorics. Related questions are motivated by computer science. Here the investigator attempts to develop applications of the differential equations method to prove good performance of randomized algorithms without an explicit solution, or numerical approximation of the solution, of the associated differential equation (the need for such information often significantly complicates application of this method).This research project is an investigation of discrete mathematical objects, like networks or codes, whose evolution over time is determined by a sequence of random choices. We are particularly interested in processes where there is dependence between the choices made in different rounds. While such processes can be good models for the dynamics of real-world phenomenon, like the spread of a disease in a network or phase transitions in materials, the focus of this research is on processes that generate interesting mathematical objects. Randomness has long played an important role in the construction of sophisticated combinatorial objects, just as randomness plays an a central role in many algorithms in computer science. This research is expected to have an impact in multiple domains as the investigator shall develop general tools for understanding the evolution of these systems over time.

这方面的研究主要集中在概率组合学领域，包括研究随机组合结构、随机算法和组合学的概率存在性证明（即概率方法）。近年来，该领域已经看到了一种技术的发展，用于证明离散随机过程的关键统计可能随着过程的发展而保持接近其预期轨迹。这种方法被称为贪心算法和随机图处理的微分方程方法。研究者最近扩展了这种技术来分析无三角形过程，这是一种“受控的”随机图过程，可以在无三角形图的空间上产生有趣的分布。研究者表明，从这种分布中绘制的图形可能具有最小可能的常数乘因子内的独立数。换句话说，无三角形过程产生拉姆齐R（3,t）图。研究者应继续发展这一方法来研究相关的“受控”随机过程，重点是从极值组合学的角度来看有趣的过程。相关的问题是由计算机科学引起的。在这里，研究者试图开发微分方程方法的应用，以证明随机算法在没有显式解或相关微分方程解的数值近似的情况下的良好性能（对此类信息的需求通常使该方法的应用变得非常复杂）。这个研究项目是对离散数学对象的调查，比如网络或代码，它们随时间的演变是由一系列随机选择决定的。我们对在不同回合中所做的选择之间存在依赖关系的过程特别感兴趣。虽然这些过程可以成为现实世界现象动力学的良好模型，比如疾病在网络中的传播或材料中的相变，但本研究的重点是产生有趣的数学对象的过程。长期以来，随机性在复杂组合对象的构建中一直发挥着重要作用，就像随机性在计算机科学的许多算法中发挥着核心作用一样。这项研究预计将在多个领域产生影响，因为研究者应该开发通用工具来理解这些系统随时间的演变。