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In this paper we extend the maximum spanning tree (MST) dependency parsing framework of McDonald et al.(2005c) to incorporate higher-order feature representations and allow dependency structures with multiple parents per word. We show that those extensions can make the MST framework computationally intractable, but that the intractability can be circumvented with new approximate parsing algorithms. We conclude with experiments showing that discriminative online learning using those approximate algorithms achieves the best reported parsing accuracy for Czech and Danish.