The complexity of finding the maximum spanning DAG and other restrictions for DAG parsing of natural language

Recently, there has been renewed interest in semantic dependency parsing, among which one of the paradigms focuses on parsing directed acyclic graphs (DAGs). Consideration of the decoding problem in natural language semantic dependency parsing as finding a maximum spanning DAG of a weighted directed graph carries many complexities. In particular, the computational complexity (and approximability) of the problem has not been addressed in the literature to date. This paper helps to fill this gap, showing that this general problem is APX-hard, and is NP-hard even under the planar restriction, in the graphtheoretic sense. On the other hand, we show that under the restriction of projectivity, the problem has a straightforward O(n) algorithm. We also give some empirical evidence of the algorithmic importance of these graph restrictions, on data from the SemEval 2014 task 8 on Broad Coverage Semantic Dependency Parsing.