Linear Algebraic Techniques in Algorithmic Graph Theory

基本图问题的算法是计算机科学和工程中的重要工具。在过去的几十年里，人们进行了广泛的研究来开发高效的算法，其中大多数是基于组合技术的。新的线性代数技术已经发展起来，在这些基本问题上取得了令人惊讶和重大的进展。我以前的研究开发了线性代数技术，为图问题设计更好的逼近算法和更快的精确算法。我计划继续这一正在进行的研究，并确定了三个有前途的研究方向。*谱图理论：应用谱技术研究组合图的性质已经有很长的历史了。经典的结果是将第二特征值与图的边展开联系起来，并用它来开发有效的图划分算法和分析随机游动的混合时间。最近，已经有一些令人兴奋的新结果将其他特征值与组合图的性质联系起来。这种新的联系为更好地分析现有算法，设计更好的近似算法，以及为各种图问题设计更快的算法开辟了道路。我相信，这种方法将为一些突出的公开问题带来新的见解，包括小集合扩展猜想、二部匹配问题和对数秩猜想。*随机游动：随机游动是图上具有广泛算法应用的简单随机过程。随机游动的一个经典应用是设计随机抽样算法。最近，随机游动的一个经典应用是设计随机抽样算法。随机游动已经被用来设计次线性时间图算法。我计划对随机游动方法进行系统的研究，设计精确的和近似的算法，并证明逼近结果的难度。此外，结合我们关于随机游动和图扩展的新知识，我想重温随机采样中的未决问题，包括拟阵展开猜想和图着色的Glauber动力学的混合时间。*线性代数算法：最近，使用线性代数算法设计基本图形问题的快速算法已经取得了重大进展。这一领域的一个主要开放问题是为加权问题设计快速代数算法。我还计划开发这种方法来解决额外的图形问题和线性代数问题，并研究图论和线性代数之间的相互作用。*我相信，实现这个建议中的目标将对计算理论以及其他领域，如机器学习、应用概率和符号计算产生重大影响。我还希望通过与我的学生学习这些新兴技术，对HQP的培训做出重大贡献。**

Algorithms for fundamental graph problems are important tools in computer science and engineering.  Extensive research has been done in the past decades to develop efficient algorithms, most of them based on combinatorial techniques.  In recent years, new linear algebraic techniques have been developed to make surprising and significant progress in these fundamental problems.  My previous research develops linear algebraic techniques to design both better approximation algorithms and faster exact algorithms for graph problems.  I plan to continue this line of ongoing research and have identified three promising directions to investigate.****Spectral graph theory: The application of spectral techniques to study combinatorial graph properties has a long history.  The classical result relates the second eigenvalue to the edge expansion of the graph, and it has been used to develop efficient algorithms for graph partitioning problems and to analyze the mixing time of random walks.  Recently, there have been exciting new results relating other eigenvalues to combinatorial graph properties.  This new connection opens up ways to do better analysis of existing algorithms, to design better approximation algorithms and to design faster algorithms for various graph problems.  I believe that this approach will bring new insights into some outstanding open problems including the small-set expansion conjecture, the bipartite matching problem, and the log-rank conjecture.****Random walks: Random walks are simple stochastic processes on graphs that have a wide range of algorithmic applications.  One classical application of random walks is in designing random sampling algorithms.  Recently, random walks have been used to design sublinear time graph algorithms.  I plan to do a systematic study of the random walks method, to design exact and approximation algorithms and to prove hardness of approximation results.  Also, with our new knowledge about random walks and graph expansion, I would like to revisit the outstanding open problems in random sampling, including the matroid expansion conjecture and the mixing time of the Glauber dynamics for graph coloring.***Linear algebraic algorithms: Recently, there has been significant progress in designing fast algorithms for fundamental graph problems using linear algebraic algorithms.  A major open problem in this area is to design fast algebraic algorithms for weighted problems.  I also plan to develop this approach to solve additional graph problems and linear algebraic problems, and to study the interplay between graph theory and linear algebra.***I believe that achieving the goals in this proposal would have major impacts in the theory of computing, as well as in other areas such as machine learning, applied probability, and symbolic computation.  I also expect to have significant contributions to the training of HQPs by acquiring these emerging techniques with my students. **