Let \(M_1=(E,\cI_1)\) and \(M_2=(E,\cI_2)\) be two matroids over the same groundset \(E\). The matroid union of \(M_1\) and \(M_2\) is a new matroid \(M_1 \lor M_2\) over the ground set \(E\) where the independent sets are the sets which can be partitioned into an independent set of \(M_1\) and an independent set of \(M_2\). More formally,

\[I \in \cI \iff I= I_1 \cup I_2 \text{ s.t. }I_1 \cap I_2 =\emptyset \text{, }I_1 \in \cI_1 \text{ and }I_2 \in \cI_2.\]

The matroid union theorem guarantees that this is indeed a matroid and that


**Distance problem on the

Diameter problem on the base polytope:

Given a matroid \(M=(E,\cI)\) find two bases \(B_1\), \(B_2\) such that $$ B_1 \Delta B_2 $$.