CAREER: Phase Transitions in Randomized Combinatorial Search and Optimization Problems

职业：随机组合搜索和优化问题中的相变

• 批准号: 1940092
• 负责人: Nike Sun
• 金额: $ 44.99万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-06-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1940092&HistoricalAwards=false
• 关键词: CAREER; Phase; Transitions; Randomized; Combinatorial

This project centers on constraint satisfaction problems (csps) - archetypal examples of combinatorial search/optimization problems - with a principal aim of building mathematical theory for these problems. A further objective is to promote connections to statistical physics and computer science, where random csps are fundamental models. Indeed, recent progress on random csps has crucially relied on the exchange of ideas among several disciplines: probability, statistical physics, combinatorics, and computer science. The proposed research will endeavor to advance this dialogue, which has potential to open new research avenues in those disciplines. The proposed research is interdisciplinary in nature: its primary focus is in the development of probability theory, but it is expected that the research approach will be much influenced by developments in statistical physics and computer science. The educational component of the proposal seeks to further promote this interdisciplinary aspect, in classroom education as well as in mentorship of graduate students. With the proliferation of statistical inference problems on large datasets, the development of fast algorithms for combinatorial problems becomes an increasingly urgent problem. At the same time it becomes more important to quantify fundamental barriers in these problems, in terms of information-theoretic and algorithmic limits. This study is complemented by proposing to develop more robust techniques to handle a wider range of problems, including some concrete problems of interest for network theory and machine learning. The educational component involves curriculum development at undergraduate and graduate levels, graduate special topic course development, and mentoring graduate students and postdocs.Combinatorial search/optimization problems are prominent in a wide range of scientific contexts. Broadly, the defining feature of these problems is that the a priori solution requires exhaustive search over a combinatorial state space such as {0,1}^n. A major challenge is that many such problems are expected to be computationally intractable in worst-case instances (np-hard). In response to this, significant attention has been directed towards random problem instances - they represent natural average-case scenarios, and serve as a useful practical benchmark. A deeper understanding of obstacles in the random setting has potential to inspire algorithmic advances. Practical considerations aside, random problem instances are of deep theoretical interest, and full of rich connections among diverse fields of research. In probability theory, random csps have contributed numerous long-standing open problems - especially ones concerning various phase transitions, conjectured either from numerical simulations or from physical heuristics. A notable problem of this type is the location of sharp satisfiability thresholds. It is a widely held belief that for many random csps, the solution space has a complicated geometric structure; and that this is precisely what obstructs standard probabilistic approaches for analyzing phase transitions. Moreover, it is conjectured that essential features of the solution space geometry are universal to a large class of random csps. Some components of this conjectural picture have been validated by recent progress in the theory of random csps, including results on satisfiability thresholds and partition function asymptotics. However, many key aspects of this picture - particularly ones relevant to algorithmic challenges - remain at the level of conjecture. One of the main goals of the current project is to shed light on these questions: to this end, specific research problems are posed regarding asymptotic properties of the solution space (in the search context) and energy landscape (in the optimization context).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目以约束满足问题（csps）为中心 - 组合搜索/优化问题的典型示例 - 其主要目的是为这些问题建立数学理论。进一步的目标是促进与统计物理学和计算机科学的联系，其中随机 csp 是基本模型。事实上，随机 csps 的最新进展很大程度上依赖于多个学科之间的思想交流：概率、统计物理、组合学和计算机科学。拟议的研究将努力推进这种对话，这有可能在这些学科中开辟新的研究途径。拟议的研究本质上是跨学科的：其主要重点是概率论的发展，但预计研究方法将受到统计物理学和计算机科学发展的很大影响。该提案的教育部分旨在进一步促进课堂教育和研究生指导中的跨学科方面。随着大型数据集统计推理问题的激增，开发组合问题的快速算法成为一个日益紧迫的问题。与此同时，在信息论和算法限制方面量化这些问题的基本障碍变得更加重要。这项研究还提出开发更强大的技术来处理更广泛的问题，包括网络理论和机器学习感兴趣的一些具体问题。教育部分涉及本科生和研究生水平的课程开发、研究生专题课程开发以及指导研究生和博士后。组合搜索/优化问题在广泛的科学背景中都很突出。从广义上讲，这些问题的定义特征是先验解决方案需要对组合状态空间（例如 {0,1}^n）进行详尽的搜索。一个主要挑战是，许多此类问题在最坏情况下（np-hard）预计将难以计算解决。为此，人们对随机问题实例给予了极大的关注——它们代表了自然的平均情况场景，并作为有用的实用基准。对随机环境中的障碍的更深入理解有可能激发算法的进步。除了实际考虑之外，随机问题实例具有深刻的理论意义，并且在不同研究领域之间充满了丰富的联系。在概率论中，随机 csps 提出了许多长期存在的开放性问题，尤其是涉及各种相变的问题，这些问题是通过数值模拟或物理启发法推测出来的。这种类型的一个值得注意的问题是尖锐可满足性阈值的位置。人们普遍认为，对于许多随机 csp，解空间具有复杂的几何结构；这正是阻碍分析相变的标准概率方法的原因。此外，据推测解空间几何的基本特征对于一大类随机 csp 是通用的。这一猜想图景的一些组成部分已被随机 csps 理论的最新进展所验证，包括可满足性阈值和配分函数渐近的结果。然而，这幅图景的许多关键方面——特别是与算法挑战相关的方面——仍然停留在猜测的水平。当前项目的主要目标之一是阐明这些问题：为此，提出了关于解空间（在搜索背景下）和能源景观（在优化背景下）的渐近性质的具体研究问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。