Post #112

Some notes on reconstruction conjecture Studying reconstruction conjecture is going down the rabbit hole 🐇 It seems so simple, but after a few steps its complexity unfolds and you find yourself reading another paper that provides some extra insights which only steer you away from the target. Formulation. You have a graph. You delete one vertex from it, and put obtained subgraph to a list, called deck. Then you return to the original graph, delete another vertex and again add obtained subgraph to the deck. And so on, for all n different vertices. The main question is whether there are two non-isomorphic graphs that have the same deck. Difficulty. I already wrote about this conjecture, showing that if you delete more than 1 vertex then there are indeed two different graphs with the same decks. That's the fact that you delete only one vertex makes it more difficult. Now think, you have n subgraphs, each has (n-1) original vertices, it seems you just have so much information as an input, O(n^3), to say everything about the original graph, which has only O(n^2). But no, it's still very hard to solve it. So what if we have a deck instead and try to recover original graph. Can we do it fast? I'm maybe wrong, but it seems there is no known polynomial-time algorithms and even the complexity of this problem is not clear. This paper, for specific type of graphs, shows an algorithm O(n^8), which is the biggest factor I've seen for poly-time algorithms. This paper studies some similar problems related to reconstruction conjecture, showing that just checking that a graph has specific deck is already equivalent to graph isomorphism. Finally, there is a manual 1991 on all major achievements to date showing that reconstruction conjecture is solved for some families of graphs such as regular graphs. Insights. After reading all this, I have the following thoughts. * It would be cool to write a paper on the algorithm for reconstruction of the graph from its deck. I have some trivial idea how to do it via graph isomorphism solver and some heuristics. Actually the problem is not purely abstract and has a lot of important applications to chemoinformatics for example, so it's interesting. * It would be cool to write a follow-up paper which uses RL to improve heuristics. Graph reconstruction is under radar compared to TSP, so it would be first-of-its-kind RL algorithm for such problem. * Initially I had the idea that if I take hard instances for graph isomorphism and compute decks for such graphs then I would get the same decks for non-isomorphic graphs, solving reconstruction conjecture. The reason why I thought this was that hard non-isomorphic graphs are very similar to each other, so their subgraphs would be identical. But, regular graphs are already proved to be reconstructible, so many hard instances won't work (such as SRG or Miyazaki graphs). Then, there are "very hard" graphs, which are half-regular (there are only two values for degrees across all nodes). I tried those, but found that decks are completely different for non-isomorphic graphs. So my idea failed and I spent two days reading, thinking, and implementing it 😀