This paper presents a novel method, called Deformable Polygonal Flow Matching (DPFM), for the generation of polygonal arrangements such as jigsaw puzzles and floor plans. DPFM is a Flow Matching framework that enables the generation process to deform, rotate, and translate poly- gons while decoupling these transformation, allowing to tog- gle them individually. Able to combine the spatial reason- ing capabilities of arrangement models with the flexibility of position-based models, it covers a wide range of appli- cations within a unified formulation, from noiseless puzzle solving using rigid alignments to unconstrained floor plans generation. We represent data using a hierarchical graph com- posed of a topological subgraph encoding connectivity infor- mation and semantics (such as room types for floor plans), and a geometrical subgraph encoding the 1D polygonal loop of each shape. DPFM also leverages Flow Matching’s arbi- trary prior distributions for geometric constraints by design- ing priors with domain knowledge. Rather than starting the generation process from uninformed distributions, the gener- ation is constrained through the informed priors since the ini- tialization stage. The qualitative and quantitative evaluations of our method, ran on the RPLAN and jigsaw puzzle datasets, demonstrate strong performance. DPFM outperforms task- specific methods, becoming the new state-of-the-art for 2D arrangement generation. As our results show, DPFM is able to solve novel tasks such as puzzle denoising, where pieces are reconstructed from noisy versions and arranged into a valid puzzle in parallel.