We will focus on some algorithmic aspects of spectral or algebraic graph theory. We will mainly discuss Laplacian systems, how to solve them, and their applications. Spielman and Teng in their seminal works:
· Spielman and Teng, Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems, Siam journal on Matrix analysis and applications, 2006.
· Spielman and Teng, A Local Clustering Algorithm for Massive Graphs and Its Application to Nearly Linear Time Graph Partitioning, Siam journal on computing 2013
· Spielman and Teng, Spectral Sparsification of Graphs, Siam journal on computing 2014.
were the first that showed how to solve Laplacian systems in near linear time. Their work has been subsequently improved and simplified in several papers, some of which we will touch in this seminar.
Our main guiding text will be the short monograph:
· Vishnoi, Lx = b, Laplacian Solvers and Their Algorithmic Application [V]
Additional good sources are
· Trevisan, Lecture Notes on Expansion, Sparsest Cut, and Spectral Graph Theory, 2016 [T]
· Kannan, Vempala, Spectral Algorithms, 2009
· Godsil, Royle, Algebraic Graph Theory, 2001, Springer (approach me to borrow a copy) [GR]
1) (Mar 3) An introductory lecture
2) (Mar 10) Colbourn, Day, Nel, Unranking and ranking spanning trees of a graph, You can also use use Section 13.2 of the book by Godsil and Royle and the lecture notes of Shayan Ovies Gharan (lectures 1+2). Amit Amram
3) (Mar 17) Graph partitioning, conductance, Fiedler's Algorithm (Chapter 5+6 of [V], 4+5 in [T]. Yossi Khayet
4) (Apr 7) Balanced cuts and computing the second eigenvalue (Chapters 7-8 of [V], Chapter 9 of [T]). Yishayahu Goodman
5) (Apr 21) Benczur and Karger, Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs, Siam journal on computing 2015
6) (April 28) Spielman and Srivastava, Graph Sparsification by Effective Resistances, Siam journal on Computing 2011 (Chapters 10, 11 in [V])
7) (May 5) Christiano, Kelner, Madry, Spielman, Teng, Electrical flows, Laplacian systems, and faster approximation of maximum flow in undirected graphs, STOC 2011 (Chapters 12 in [V])
8) (May 12) Iterative methods: Gradient descent and conjugate gradient (Chapters 15,16 in [V])
9) (May 19) Preconditioning, tree preconditioners (Chapters 17,13 in [V])
11) (June 2) Koutis, Miller, and Peng, Approaching Optimality for Solving SDD Linear Systems, Siam journal on computing, 2014 (Chapters 18 in [V])
12) (June 9) Kelner, Orecchia, Sidford, Zhu, A Simple, Combinatorial Algorithm for Solving SDD Systems in Nearly-Linear Time (part 1) (Chapters 14 in [V])
13) (June 16) A Simple, Combinatorial Algorithm for Solving SDD Systems in Nearly-Linear Time (part 2)