CAREER: Phase Transitions in Randomized Combinatorial Search and Optimization Problems

这个项目集中在约束满足问题（csps）-组合搜索/优化问题的典型例子-与这些问题建立数学理论的主要目标。进一步的目标是促进与统计物理学和计算机科学的联系，随机csp是基本模型。事实上，最近在随机csp上的进展主要依赖于几个学科之间的思想交流：概率论，统计物理学，组合学和计算机科学。拟议的研究将奋进推进这一对话，这有可能在这些学科开辟新的研究途径。拟议的研究是跨学科的性质：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导师制中的这一跨学科方面。随着大数据集上统计推断问题的激增，开发组合问题的快速算法成为一个日益迫切的问题。与此同时，量化这些问题中的基本障碍变得更加重要，包括信息理论和算法限制。这项研究的补充是，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。教育部分包括本科生和研究生水平的课程开发，研究生专题课程开发，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下是突出的。广义地说，这些问题的定义特征是先验解需要在组合状态空间（如{0，1}^n）上进行穷举搜索。一个主要的挑战是，许多这样的问题预计将在计算上棘手的最坏情况下（np-hard）。针对这一点，大量的注意力已经转向随机问题实例-它们代表自然的平均情况下的情况下，并作为一个有用的实际基准。对随机环境中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，并且在不同的研究领域之间充满了丰富的联系。在概率论中，随机csps贡献了许多长期存在的开放问题-特别是那些关于各种相变的问题，无论是从数值模拟还是从物理学中得到的。这种类型的一个值得注意的问题是尖锐的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;而这正是阻碍标准概率方法分析相变的原因。此外，我们还证明了解空间几何的基本特征对一大类随机csp是普适的。这幅图的某些组成部分已经被随机csp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性的结果。然而，这一图景的许多关键方面--特别是与算法挑战相关的方面--仍然处于猜测的水平。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（在搜索环境中）和能源景观（在优化环境中）的渐近性质的具体研究问题。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.