# Matroid diameter v/s shortest path

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,

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

**Theorem:**

**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 $$ | B_1 \Delta B_2 | $$. |