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 + epsilon for any epsilon > 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 [V. Guruswami. C. Umans, S.P. Vadhan, Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes, in: IEEE Conference on Computational Complexity, IEEE Computer Society, 2007, pp. 96-108] can be used to obtain a code with these properties. We also show that the parameterized matroid-greedoid partition problem is fixed-parameter tractable. (C) 2008 Elsevier B.V. All rights reserved.