We propose a novel graph-based approach for semantic parsing that resolves two problems observed in the literature: (1) seq2seq models fail on compositional generalization tasks; (2) previous work using phrase structure parsers cannot cover all the semantic parses observed in treebanks. We prove that both MAP inference and latent tag anchoring (required for weakly-supervised learning) are NP-hard problems. We propose two optimization algorithms based on constraint smoothing and conditional gradient to approximately solve these inference problems. Experimentally, our approach delivers state-of-the-art results on GeoQuery, Scan, and Clevr, both for i.i.d. splits and for splits that test for compositional generalization.