CS614. Advanced Algorithms. L03 Quiz.

CS614. Advanced Algorithms.

L03 Quiz

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Problem 1. Partition Matroid

Show that the exchange axiom holds for the Partition Matroid defined in class.

Problem 2. Representing the Graphic Matroid

The graphic matroid of a graph $G$ can be represented by the following matrix: we have one row for each vertex, and one column for each edge. The column for edge $e$ has $+1$ in the row for one endpoint, $-1$ in the row for the other endpoint, and $0$ elsewhere; the choice of which endpoint to give which sign is arbitrary.

Argue that this is a valid representation (i.e, that the forests correspond to linearly independent columns and the subsets of edges that have cycles in them correspond to dependent columns).