1 Determining strong base-orderability of a matroid

https://mathoverflow.net/questions/194006/determining-strong-base-orderability-of-a-matroid

determining base-orderability of a matroid can be done through finding perfect matching in a bipartite graph with $\Vert E{\Vert }^2$ edges. call oracle $\Vert E{\Vert }^2$ times to build the graph and run a matching algorithm.

Is the union of strongly base-orderable matroids strongly base-orderable? yes. https://mathoverflow.net/questions/201101/is-the-union-of-strongly-base-orderable-matroids-strongly-base-orderable also see wikipedia

1.0.1 difference between base-orderable and strong base orderable?

https://arxiv.org/abs/1507.05521

an induction on the size of the set $X$

• 1-BO
• 2-BO
• …
• $\lceil r/2\rceil$-BO = SBO

given 2 bases of a matroid and a k-BO bijection f, find subsets with k+1 elements which do satisfy $k+1$-BO?

1-BO -> 2-BO is already hard…