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，人们进行了大量的研究来开发高效的算法，其中大多数基于组合技术。近年来，新的线性代数技术得到了发展，在这些基本问题上取得了惊人的重大进展。我以前的研究发展了线性代数技术，为图问题设计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。

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.