So, the REU is nearing its end for me and I have not been reporting anything here... what's the matter?
The problem is at first I got very excited, to later realize that in fact I'm stuck.
The past four weeks I was working on a coloring algorithm for shrub-depth, trying to generalize the very simple and elegant ILP solution presented by Michael Lampis in the original ND paper. I and Vojta investigated a few approaches during our time at Princeton. I decided to pursue one that looked promising, and for some time felt I solved the problem. However, when I tried writing down the solution, I found out I've only solved a part of the problem, and the easier one at that.
Roughly speaking the task is the following. I need to find a "canonical" way to describe colorings of graphs of bounded shrub-depth such that this canonical way would be expressible as a solution to an ILP, but it has to be succinct enough to only need \(f(k,d)\) variables, for \(k\) the number of labels and \(d\) the shrub-depth. This "canonical form" approach was successful in my thesis and other ND-related research, but I know of no examples where it would work for shrub-depth. We'll have to see. For now I'm in the "stuck" phase, which just sometimes happens in research. It sucks, but it is necessary.