SODA Day 2 was the official day for bidimensional dimensionality reduction in spaces of bounded doubling dimension (ok that almost made sense). Here are some highlights:
- I enjoyed Bidimensionality and EPTAS not because of the results themselves (which are interesting) but because the talk was great, and I learnt a lot about the basic framework of bidimensionality. The talk was low on technical details, but the main contribution of the paper is a stronger result connecting bidimensionality of a graph property to the existence of an (E)fficient PTAS for the problem. Earlier work needed a constant factor approximation algorithm to bootstrap the process, and they were able to relax that assumption.
- In the same session, Known Algorithms on Graphs on Bounded Treewidth are Probably Optimal was also cool. They showed that any progress on improving the exponential dependence on treewidth for a number of parametrized problems (for example from 2^tw to (2-eps)^tw) would break the Strong Exponential Time Hypothesis (i.e that SAT cannot be solved in (2-delta)^n time).
- A paper that drew a lot of attention (as seen by the huge influx of people into the room) was the work by Daskalakis and Papadimitriou (Christos gave a really great talk) on Continuous Local Search. In brief, they constructed a complexity class that combines the approximate fixpoints of PPAD and the local search of PLS and contains some natural problems from algorithmic game theory as well as numerical analysis. Indeed, gradient search can be viewed as a special example of elements of this class.
- There was a whole session on approximate geometry, and doubling dimension played a key role in constructing efficient spanners, and in getting below the JL limit. Sandwiched between these was a local JL embeddability result, the idea being that if you only care about preserving distances in your (k)-neighborhood, you can get dimensionality reduction in terms of k, rather than n. The proof itself uses a variant of the Nash embedding theorem.