Papers.

1. Parallel approximation of non-interactive zero-sum quantum games, Rahul Jain , John Watrous Electronic edition.

   Matrix mult update rule applied to quantum games.

2. Julia Kempe, Oded Regev,Ben Toner. The Unique Games Conjecture with Entangled Provers is False. Electronic Edition

   semidefinite programming should have a basic quantum interpretation. Here is a starting point.

3. Manfred Warmuth. Winnow for low rank matrices. Electronic Edition.

   Matrix multiplicative update rule. learning a hidden low rank projection matrix.

4. Sanjeev Arora, Satyen Kale: A combinatorial, primal-dual approach to semidefinite programs. STOC 2007: 227-236 Electronic Edition.

   Matrix exponentials and expander flows.

5. J. Leskovec, K. J. Lang, A. Dasgupta, and M. W. Mahoney, Statistical properties of community structure in large social and information networks, Proc. 17-th International WWW/, 695-704 (2008) Electronic Edition

   Can you come up with a natural graph model that obeys the conductance versus n plot.

6. Joshua Batson, Daniel A. Spielman, Nikhil Srivistava, Twice-Ramanujan Sparsifiers CoRR abs/0808.0163: (2008)
   

   This paper shows how to find for any graph $G$ an $O(n)$ edge graph $H$ that has approximately the same spectral norm. The bounds are impressively tight and the algorithm is polynomial time. This can be compared to the "near" linear time algorithm of Spielman that produced an $O(n \log n)$ edge graph and Srivistav that we presented in class. A possible open question is a faster algorithm with the better results.

7. Jacob Abernethy, Elad Hazan, Alexander Rakhlin, Competing in the Dark: An Efficient Algorithm for Bandit Linear Optimization. COLT 2008: 263-274; Electronic Edition
   

   This paper uses ideas from optimization (particulary regarding the potential functions used in interior point methods) to improve results in the experts framework.

8. Samuel I. Daitch, Daniel A. Spielman: Faster Approximate Lossy Generalized Flow via Interior Point Algorithms CoRR abs/0803.0988: (2008)
   

   This paper uses the Spielman-Teng iterative method (from class) for solution of linear equations and an interior point method to give new algorithms for some flow problems. Note that, for example, the min cost flow solution that they have surpasses combinatorial algorithms that were the product of decades of work. This paper has numerous ideas and will take some perserverance to understand. Please read in a group. Moreover, the instructors are happy to provide support.

9. David Tolliver, Gary L. Miller: Graph Partitioning by Spectral Rounding: Applications in Image Segmentation and Clustering. CVPR (1) 2006: 1053-1060, Electronic Edition

   This is a different rounding method for spectral partitioning. What does this do, exactly?

10. Detecting Sharp Drops in PageRank and a Simplified Local Partitioning Algorithm, EE and Local Partitioning for Directed Graphs using PageRank, EE

    These show how to use local random walks to find sparse cuts around an input sets of vertices in very fast time (i.e. proportional to the size of the cut found).

11. "Four proofs for the Cheeger inequality and graph partitioning algorithms" EE, The Lovasz-Simonovits method yields a different approach to proving Cheeger-like bounds on the relation between mixing and conductance. There are some good notes by Spielman on his webpage. This paper by Fan Chung shows how the Lovasz-Simonovits method can be applied to a bunch of walks including heat kernel (i.e. diffusion) and PageRank:

More questions.

1. What is the relationship between the multiplicative weights algorithm and the interior point algorithm for Linear Programs? The former can often be faster in terms of the number of discrete objects (for example, does not require the solution of a set of linear equations) but gets a solution that is within $1+\epsilon$ of optimal in time proportional to $O(1/\epsilon^2)$ where the latter converges in time proportional to $O(\log (1/\epsilon))$. That is, at least in terms of error, interior point is exponentially better.
2. Are there other applications of the linear solver of Spielman and Teng? Perhaps more general linear programs, perhaps using their ideas directly.
3. There is a sparsest cut algorithm that yields an approximation of $O(\sqrt{\log n})$ in $\tilde{O} (n^2)$ time. There is also a couple that run in "max flow" time of $\tilde {O} (n^{3/2})$ that yields an approximation of $O(\log n)$. Can one find one that matches the best time and the best approximation?
4. An improved analysis or simplified version of the Spielman, Teng algorithm for solving systems might be interesting. This algorithm may indeed be an amazing breakthrough. Any understanding or simplification might be interesting.
5. What do extremal examples look like for semidefinite programs? For example, what is a extremal example for the three coloring problem. Are there graphs that are not at all three colorable that could have a feasible embedding? There is work on this regarding extremal examples for MAX cut. There is also quite a bit work on this regarding extremal examples with respect to generalizations of semidefinite programs and linear programs.