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Graph-based methods, which decompose the score of a dependency tree intoscores of dependency arcs, are popular in dependency parsing for decades.Recently, \citet{Yang2022Span} propose a headed-span-based method thatdecomposes the score of a dependency tree into scores of headed spans. Theyshow improvement over first-order graph-based methods. However, their methoddoes not score dependency arcs at all, and dependency arcs are implicitlyinduced by their cubic-time algorithm, which is possibly sub-optimal sincemodeling dependency arcs is intuitively useful. In this work, we aim to combinegraph-based and headed-span-based methods, incorporating both arc scores andheaded span scores into our model. First, we show a direct way to combine with$O(n^4)$ parsing complexity. To decrease complexity, inspired by the classicalhead-splitting trick, we show two $O(n^3)$ dynamic programming algorithms tocombine first- and second-order graph-based and headed-span-based methods. Ourexperiments on PTB, CTB, and UD show that combining first-order graph-based andheaded-span-based methods is effective. We also confirm the effectiveness ofsecond-order graph-based parsing in the deep learning age, however, we observemarginal or no improvement when combining second-order graph-based andheaded-span-based methods. Our code is publicly available at\url{https://github.com/sustcsonglin/span-based-dependency-parsing}.

Combining (second-order) graph-based and headed-span-based projective dependency parsing