# Hierarchical Forecast Reconciliation on Networks: A Network Flow Optimization Formulation

arXiv: https://arxiv.org/abs/2505.03955 · HTML: https://mlsanity.com/paper/2505.03955/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 61 references that are in the map.
- First posted on arXiv 6 May 2025.

## Builds on (most cited first, 24 of 24)

- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [N-BEATS: Neural basis expansion analysis for interpretable time series forecasting](https://mlsanity.com/paper/1905.10437/) (2019)
- [Long Short-Term Memory Based Recurrent Neural Network Architectures for Large Vocabulary Speech Recognition](https://mlsanity.com/paper/1402.1128/) (2014)
- [Clarabel: An interior-point solver for conic programs with quadratic objectives](https://mlsanity.com/paper/2405.12762/) (2024)
- [Maximum Flow and Minimum-Cost Flow in Almost-Linear Time](https://mlsanity.com/paper/2203.00671/) (2022)
- [Predictive Flows for Faster Ford-Fulkerson](https://mlsanity.com/paper/2303.00837/) (2023)
- [Probabilistic Hierarchical Forecasting with Deep Poisson Mixtures](https://mlsanity.com/paper/2110.13179/) (2021)
- [Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs](https://mlsanity.com/paper/2009.01802/) (2020)
- [Learning-Augmented Maximum Flow](https://mlsanity.com/paper/2207.12911/) (2022)
- [Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back](https://mlsanity.com/paper/1307.2205/) (2013)
- [Computing Maximum Flow with Augmenting Electrical Flows](https://mlsanity.com/paper/1608.06016/) (2016)
- [Recent Advances in Fully Dynamic Graph Algorithms](https://mlsanity.com/paper/2102.11169/) (2021)
- [Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, s-t Shortest Path, and Minimum-Cost Flow](https://mlsanity.com/paper/2311.18295/) (2023)
- [Dirichlet Proportions Model for Hierarchically Coherent Probabilistic Forecasting](https://mlsanity.com/paper/2204.10414/) (2022)
- [Nested Dissection Meets IPMs: Planar Min-Cost Flow in Nearly-Linear Time](https://mlsanity.com/paper/2205.01562/) (2022)
- [A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow](https://mlsanity.com/paper/2309.16629/) (2023)
- [Sublinear-Time Decremental Algorithms for Single-Source Reachability and Shortest Paths on Directed Graphs](https://mlsanity.com/paper/1504.07959/) (2015)
- [A Simpler and Faster Strongly Polynomial Algorithm for Generalized Flow Maximization](https://mlsanity.com/paper/1611.01778/) (2016)
- [Improved Algorithms for Decremental Single-Source Reachability on Directed Graphs](https://mlsanity.com/paper/1612.03856/) (2016)
- [Warm-starting Push-Relabel](https://mlsanity.com/paper/2405.18568/) (2024)
- [New Algorithms and Lower Bounds for All-Pairs Max-Flow in Undirected Graphs](https://mlsanity.com/paper/1901.01412/) (2019)
- [Faster High Accuracy Multi-Commodity Flow from Single-Commodity Techniques](https://mlsanity.com/paper/2304.12992/) (2023)
- [Incremental Approximate Maximum Flow on Undirected Graphs in Subpolynomial Update Time](https://mlsanity.com/paper/2311.03174/) (2023)

## Abstract (arXiv)

Hierarchical forecasting with reconciliation requires forecasting values of a hierarchy (e.g.~customer demand in a state and district), such that forecast values are linked (e.g.~ district forecasts should add up to the state forecast). Basic forecasting provides no guarantee for these desired structural relationships. Reconciliation addresses this problem, which is crucial for organizations requiring coherent predictions across multiple aggregation levels. Current methods like minimum trace (MinT) are mostly limited to tree structures and are computationally expensive. We introduce FlowRec, which reformulates hierarchical forecast reconciliation as a network flow optimization, enabling forecasting on generalized network structures. While reconciliation under the ℓ0 norm is NP-hard, we prove polynomial-time solvability for all ℓp > 0 norms and , for any strictly convex and continuously differentiable loss function. For sparse networks, FlowRec achieves O(n2log n) complexity, significantly improving upon MinT's O(n3). Furthermore, we prove that FlowRec extends MinT to handle general networks, replacing MinT's error-covariance estimation step with direct network structural information. A key novelty of our approach is its handling of dynamic scenarios: while traditional methods recompute both base forecasts and reconciliation, FlowRec provides efficient localised updates with optimality guarantees. Monotonicity ensures that when forecasts improve incrementally, the initial reconciliation remains optimal. We also establish efficient, error-bounded approximate reconciliation, enabling fast updates in time-critical applications. Experiments on both simulated and real benchmarks demonstrate that FlowRec improves accuracy, runtime by 3-40x and memory usage by 5-7x. These results establish FlowRec as a powerful tool for large-scale hierarchical forecasting applications.