- Ankur Moitra and Tom Leighton solved the greedy embedding conjecture for 3-connected planar graphs, posed by Papadimitriou and Ratajczak in 2004. A path in the plane is distance-decreasing if the distance to one endpoint is monotonically decreasing as we walk along the path from the other endpoint. An embedding of a graph in the plane is greedy if every pair of nodes in the graph is connected by a distance-decreasing path. Papadimitriou and Ratajczak conjectured that every 3-connected planar graph has a greedy embedding. 3-connected planar graphs have tons of lovely geometric properties via theorems of Cauchy, Tutte, Steinitz, Koebe-Andreev-Thurston, and others, none of which proved useful in Moitra and Leighton's solution! Instead, they argue that any 3-connected planar graph (in fact, any circuit graph) has a spanning Christmas cactus, a subgraph in which every edge belongs to at most one cycle and deleting any vertex leaves at most two components, and then show how to greedily embed any Christmas cactus.
- Tasos Sidiropoulos described inapproximability resuls for metric embeddings into R^d, which he developed with Jiri Matousek. Given an n-point metric space M and a target dimension d, it is natural to ask for the minimum-distortion embedding of M into Euclidean d-space. Unfortunately, even for the case d=1, approximating the minimum distortion to small polynomial factors is NP-hard. Tasos and Jirka bootstrapped this known result to arbitrary dimensions, using a lovely product construction. If M is a bad example for embeddng into the line, then MxS is a bad example for embedding into R^d, where S is a sufficiently dense net on the (d-1)-sphere. Each copy of S must be embedded on an approximate sphere. A clever cohomology argument (hooray for Poincaré-Alexander duality!) then implies that these quasi-spheres are properly nested, which implies that the distortion is at least as bad as the distortion of M into the line. Thus result implies that random projection not only gives near-optimal worst-case distortion, but it also gives a nearly optimal approximation of the distortion for any metric space.
On our way to dinner last night, Mihai Patrascu asked me if the stories about my faculty job interview at MIT were true. Apparently, I impressed somebody (Mihai wouldn't say who) enough to become a cautionary example to MIT theory students. The story has become a little distorted over the last ten years, but yes, the essential details are accurate.
For any students entering the academic job market, who desperately want to avoid an offer from MIT, let me offer the following script, which proved wildly successful for me:
- Gerry Sussman:
- What's going to be the most important development in computer science in the next 20 years?
- Yours truly:
- I don't know, and neither does anyone else!
I'm not crazy enough to think that was the only reason MIT didn't hire me—truthfully, I think they made the right decision—but it certainly didn't help. On the other hand, aside from a few added details about high school students and pornography, I'm not sure my answer would be much different today.