Mastodawn

I think I solved it! https://codeberg.org/fogti/public-docs/raw/branch/main/NWC-related/matroid-circuits/mc-resolve.pdf

"Resolving matroid circuit exchanges", as part of my effort to pin down what actually makes the Neil-White conjecture complicated.

I unfortunately had to relax the conditions for triangle generators to exist. But this shouldn't hamper the usefulness too much.

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Έλλεν Εμίλια Ά.ζ.Mar 24

And to make this easily demonstrable, I implemented the algorithm in Haskell, albeit I haven't yet been able to test it enough to make sure it properly works for all cases (in particular, the single test case that's in the repo currently only tests SymmetricExchanges).
https://codeberg.org/fogti/nwc-decomb

#matroid #matroids

nwc-decomb

Neil White conjecture instance decombinator.

Codeberg.org

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Έλλεν Εμίλια Ά.ζ.Mar 24

Funnily enough, this does afaik basically solve a weird "twist" on the Neil White conjecture (https://en.wikipedia.org/w/index.php?title=Basis_of_a_matroid&oldid=1320304207#Neil_White's_conjecture), where we only manage to reduce the amount of exchanges inductively (meaning the list of all necessary exchanges, including symmetric exchanges and triangle / 3-cyclic exchanges, is finite), but can't guarantee that the originally supplied list of bases suffices.

Basis of a matroid - Wikipedia

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Έλλεν Εμίλια Ά.ζ.

And triangle exchanges are a bit weird because they're necessary to deal with uniform matroid minors, where one normally would think that uniform matroids are particularly easy to deal with.