Linear Algebraic Techniques in Algorithmic Graph Theory

基本图问题的算法是计算机科学与工程中的重要工具。 在过去的几十年里，人们进行了广泛的研究来开发有效的算法，其中大多数是基于组合技术。 近年来，新的线性代数技术已经发展，使这些基本问题的令人惊讶的和重大的进展。 我以前的研究开发了线性代数技术，为图形问题设计了更好的近似算法和更快的精确算法。 我计划继续这条正在进行的研究路线，并确定了三个有希望的研究方向。 谱图理论：应用谱技术来研究组合图的性质有很长的历史。 经典的结果与第二个特征值的边缘扩张的图，它已被用来开发有效的算法，图划分问题和分析的混合时间的随机游动。 最近，已经有令人兴奋的新结果，其他特征值的组合图形属性。 这种新的连接开辟了更好地分析现有算法的方法，设计更好的近似算法，并为各种图形问题设计更快的算法。 我相信，这种方法将带来新的见解，一些突出的开放问题，包括小集膨胀猜想，二分匹配问题，对数秩猜想。 随机游走：随机游走是图上的简单随机过程，具有广泛的算法应用。 随机游动的一个经典应用是设计随机抽样算法。 最近，随机游动已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，设计精确算法和近似算法，并证明近似结果的困难性。 此外，随着我们的新知识，随机游动和图的扩展，我想重温随机抽样中的悬而未决的开放问题，包括拟阵扩展猜想和混合时间的Glauber动态图着色。 线性代数算法：近年来，在利用线性代数算法设计基本图问题的快速算法方面取得了重大进展。 在这一领域的一个主要的开放问题是设计快速的代数算法加权问题。 我还计划发展这种方法来解决更多的图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，实现这一提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。 我也希望通过与我的学生一起获得这些新兴技术，为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.