The complexity of the matroid-greedoid partition problem

We show that the maximum matroid-greedoid partition problem is NP-hard to approximate to within 1/2 + ε for any ε > 0, which matches the trivial factor 1/2 approximation algorithm. The main tool in our hardness of approximation result is an extractor code with polynomial rate, alphabet size and list size, together with an efficient algorithm for list-decoding. We show that the recent extractor construction of Guruswami, Umans and Vadhan [5] can be used to obtain a code with these properties. We also show that the parameterized matroid-greedoid partition problem is fixed-parameter tractable.