Finding the smallest binarization of a CFG is NP-hard

Grammar binarization is the process and result of transforming a grammar to an equivalent form whose rules contain at most two symbols in their right-hand side. Binarization is used, explicitly or implicity, by a wide range of parsers for context-free grammars and other grammatical formalisms. Non-trivial grammars can be binarized in multiple ways, but in order to optimize the parser's computational cost, it is convenient to choose a binarization that is as small as possible. While several authors have explored heuristics to obtain compact binarizations, none of them guarantee that the resulting grammar has minimum size. However, to our knowledge, no hardness results for this problem have been published. In this article, we address this issue and prove that the problem of finding a minimum binarization of a given context-free grammar is NP-hard, by reduction from vertex cover. We also provide a lower bound on the approximability of this problem.