From file: lecture1.html

    1. There have been many proofs that were claimed to be correct for the four color theorem. Well.. Is this one correct? I certainly can't tell at this point. So, (attempt to) shed some light.
    2. This is an approach to computer verification of mathematical proofs. One such approach has lent credence to the four color theorem's proof. If you are into verification, perhaps you would like to make an excelle presentation of this.

From file: lecture2.html

  1. VLSI layout. Perhaps compare simple approaches to VLSI layout. For example, local improvement versus finding partitions and recursing versus laying out by the first two eigenvectors. I would do a "simplified layout" where one simply places nodes on a grid and attempts to minimize the total wirelength. Another complication is that in VLSI, one actually routes "nets", i.e., hyperedges that consist of more than two "endpoints" that should be connected perhaps by a Steiner tree (which again is difficult.) I can give some suggestions or not. (One suspects that local improvement should be done afterwards on any approach.) Try it on the networks at Charles Alpert's Benchmark. Or if you want to restrict yourself to graphs, try these
  2. Theoretical. A generalization of planarity is the notion of minor exclusion. That is, a planar graph excludes $K_5$. There is a separator theorem for any graph that excludes $K_h$ where the size is $O(h^{3/2} \sqrt{n})$ by Alon, Seymour, Thomas. . One can modify this construction as suggested here and get $O(h \sqrt{n \log n}).$ Can you get $O(h \sqrt{n})$? Warning: this may not be too easy, but understanding the previous is fine as well.
  3. There is a lovely proof of the planar separator theorem using maximum flow, understand and give a presentation of this proof. Speculate on extending this to higher genus surfaces.

From file: lecture3.html

    1. Try it! Use the eigenvalue method and compare it for finding sparse cuts in these. You can compare to METIS. Probably, you will get killed (and be slow) compared to METIS. Explain why.
    2. Evilness. Construct a family of examples that makes an eigenvalue approach or METIS perform extremely badly (is not within any constant factor of optimal.) Generate them, and show me. Try to understand why.
    3. Theory. Read and understand this paper. Perhaps make some progress on the main conjecture.
    4. Understand and extend, experiment with ideas in this paper on rounding spectral cuts.

From file: lecture4.html

  1. See lecture3.

From file: lecture5.html

  1. See lecture3.

From file: lecture6.html

  1. Read and understand and present this paper. There is a followup paper by Hagerup that is perhaps easier to understand.
  2. Understand and present the state of the art in predecessors. Perhaps start here.

From file: lecture7.html

  1. Probably impossible but improve the following here. If no improvement, you fail. Well, just kidding, make a presentation of it and its predecessor (Karger, Benzcur.)

From file: lecture8.html

  1. Make a presentation of $O(\log n)$ amortized complexity implementation of dynamic trees.

From file: lecture9.html

  1. Use the flow based algorithm to write code for image segmentation. Compare to previous results. Perhaps start here to get graphs of appropriate edge weights from various images and for comparison.
  2. Read, understand, make a presentation of Orlin's strongly polynomial time algorithm for min cost flow.

From file: lecture10.html

    1. Present a survey on the state of the art regarding the Hirsch conjecture (the diameter of the simplex is $n-d.$).

From file: lecture12.html

    1. Use these ideas in lecture 12 to code a VLSI routing algorithm to minimize the maximum congestion in any track. Extra points for dealing with multiway nets. Again, see Charles Alpert's Benchmark for examples.

From file: lecture13.html

  1. Understand and discuss whether the $\sqrt{n}$ iterations per drop in potential is necessary. What makes it necessary. Perhaps, start here where they suggest $n^{.25}$ may happen reasonably often. My intuition is that the bad case for the primal step and the bad case for the dual may be better than what it seems. That is, the primal only makes a small drop if there is a single large dimension which restricts how far you can go. Perhaps a different direction is ok. Similarly, the Cauchy-Swartz limit in the dual step seems good when there are a few large directions. Do I have a sign error?
  2. Can you explain why this algorithm gets exponential convergence where the experts algorithm gets polynomial convergence for two person zero sum games. That is the latter takes $O(1/\epsilon^2)$ iterations to get within $(1+\epsilon)$ whereas the former takes $O(\log (1/\epsilon))$ along with various dependencies on $n$ and $m$. See the surveys by Goemans and Arora on the two methods on the course webpage.