CAREER: Phase Transitions in Randomized Combinatorial Search and Optimization Problems

本项目以约束满足问题（csps）为中心——组合搜索/优化问题的典型例子——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csp的最新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。拟议的研究将努力推动这种对话，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。该提案的教育部分寻求在课堂教育和研究生指导中进一步促进这种跨学科方面。随着大型数据集上统计推理问题的激增，开发快速的组合问题算法成为一个日益迫切的问题。同时，从信息论和算法限制的角度来量化这些问题中的基本障碍变得更加重要。本研究还建议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。教育部分包括本科和研究生阶段的课程开发，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下是突出的。广义地说，这些问题的定义特征是先验解决方案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在最坏情况下难以计算（np-hard）。为了回应这一点，人们将大量的注意力集中在随机问题实例上——它们代表了自然的平均情况，并作为有用的实际基准。对随机环境中障碍的更深入理解有可能激发算法的进步。撇开实际考虑不谈，随机问题实例具有深刻的理论意义，并且在各个研究领域之间充满了丰富的联系。在概率论中，随机csp产生了许多长期存在的开放问题，特别是那些涉及各种相变的问题，这些问题要么是从数值模拟中推测出来的，要么是从物理启发式中推断出来的。这种类型的一个值得注意的问题是尖锐的满意阈值的位置。人们普遍认为，对于许多随机csps，解空间具有复杂的几何结构；而这正是阻碍分析相变的标准概率方法的原因。此外，我们还推测解空间几何的基本特征对于一大类随机csp具有普适性。这一猜想图景的某些组成部分已被随机csps理论的最新进展所证实，包括关于可满足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——尤其是与算法挑战相关的方面——仍处于猜测的水平。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（在搜索上下文中）和能量景观（在优化上下文中）的渐近性质的具体研究问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.