Min-Cost Flow Routing for Evidence Assembly in Long Multimodal Documents

Answering questions about long multimodal documents requires distributing a fixed evidence budget across relevant facets in text, tables, figures, and slides while avoiding near-duplicates. We present \flowreader, which formulates evidence selection as a single minimum-cost flow problem with capacity limits over a multimodal content graph. Spectral decomposition identifies latent aspects of query-relevant content and allocates the budget among them in proportion to their spectral energy. These capacity limits enforce aspect coverage during routing without requiring a language-model planning call. Query-conditioned costs prioritize chains of relevant, mutually consistent evidence. Decomposing the optimal flow produces short evidence chains, which a vision-language model reads in parallel and a reasoner reconciles. On VisDoMBench with Qwen3-VL-32B, \flowreader achieves the highest macro accuracy ($68.9$), surpassing the strongest prior system by $2.7$ points, leading on three of five subsets and attaining the highest worst-subset accuracy. It uses a measured $17.5$ content nodes per query and maintains its lead at $12.9$. Ablation studies with a fixed graph, scorer, reader, and judge show that cost design drives accuracy, capacity limits preserve it while using about three-quarters of the reader tokens required by shortest-path routing without these limits on the same network, and spectral aspects align with LLM-generated sub-questions without a planning call.