in Combinatorial Optimization April 8 , 2004 Lecture 16 Lecturer : Michel

We now consider some examples of jump systems. Example 1. Let M be a matroid over a set S, |S| = n. We let each coordinate of Z correspond to an element of S, and let J ⊆ {0, 1}n be the set of characteristic functions for bases of M , J = {χB|B a basis of M}. Claim 1. The set J is a jump system. Proof. Let x and y be the respective characteristic vectors of two bases b1 and b2 of M . A step x from x to y corresponds to either: 1. Adding to b1 an element of b2 \ b1, or 2. Removing from b1 some element of b1 \ b2. ′ / Since all bases have the same size, the set corresponding to x will never be a basis, so x′ ∈ J . We thus require there to be some step x′′ from x to y such that x′′ ∈ J , i.e., such that the set corresponding to x′′ is a basis. In both cases, this is guaranteed by Basis Exchange (see lecture 11).