**Authors:** Yue Wu, Binghong Li, Kathleen Chang, Shruti Puri, Lin Zhong Fast and accurate quantum error correction (QEC) decoding is crucial for scalable fault-tolerant quantum computation. Most-Likely-Error (MLE) decoding, while being near-optimal, is intractable on general quantum Low-Density Parity-Check (qLDPC) codes and typically relies on approximation and heuristics. We propose HyperBlossom, a unified framework that formulates MLE decoding as a Minimum-Weight Parity Factor (MWPF) problem and generalizes the blossom algorithm to hypergraphs via a similar primal-dual linear programming model with certifiable proximity bounds. HyperBlossom unifies all the existing graph-based decoders like (Hypergraph) Union-Find decoders and Minimum-Weight Perfect Matching (MWPM) decoder, thus bridging the gap between heuristic and certifying decoders. We implement HyperBlossom in software, namely Hyperion. Hyperion achieves a 4.8x lower logical error rate compared to the MWPM decoder on the distance-11 surface code and 1.6x lower logical error rate compared to a fine-tuned BPOSD decoder on the $[[90, 8, 10]]$ bivariate bicycle code under code-capacity noise. It also achieves an almost-linear average runtime scaling on both the surface code and the color code, with numerical results up to sufficiently large code distances of 99 and 31 for code-capacity noise and circuit-level noise, respectively.