What happened to cographic regularity?

Stream: Seymour

[Tristan Figueroa-Reid]

(Title) Is there a new approach for this?

[Martin Dvořák]

Ivan was not interested in working on it, so we decided to exclude it from the first phase of the project.

As for me, even if StandardRepr.dual_isBase_iff got proved somehow, I would be clueless about Matroid.IsRegular.dual because of the missing finiteness assumptions. Do you have an idea? We can bring it back if you want to work on it.

[Tristan Figueroa-Reid]

I don't think I would have anything new to bring in that regard either - I could try to work on it and see, though.

(Also, what are the phases to this project?)

[Martin Dvořák]

In the first phase, we want to prove Matroid.Is1sumOf.isRegular and Matroid.Is2sumOf.isRegular and Matroid.Is3sumOf.isRegular and write a paper about it. The first two have been finished today; as for the last one, a lot of work is still pending.

[Martin Dvořák]

Tristan Figueroa-Reid said:

I don't think I would have anything new to bring in that regard either - I could try to work on it and see, though.

If you want to work on cographicness, one piece of refactoring I can suggest to do, which will be useful for the aforementioned lemma, is to take VectorMatroid.exists_standardRepr_isBase_isTotallyUnimodular and change it so that TUness is optional; if the input is totally unimodular, the output will be totally unimodular.

A second option, probably harder, would be to reconsider https://github.com/Ivan-Sergeyev/seymour/issues/81 and https://github.com/Ivan-Sergeyev/seymour/issues/82 and have two orthogonal lemmas; one that preserves TUness but requires finiteness, the other works in the infinite case but doesn't promise that TUness is preserved. I am not entirely sure it is viable.

A third option, arguably much harder, would be to generalize VectorMatroid.exists_standardRepr_isBase_isTotallyUnimodular to not require finiteness and then do the first suggested refactor. That said, I still wouldn't know how to resolve the other problems required for Matroid.IsCographic.isRegular being proved.

As for me, I would like to focus on the 3-sum now.