# Nested Dissection Meets IPMs: Planar Min-Cost Flow in Nearly-Linear Time

arXiv: https://arxiv.org/abs/2205.01562 · HTML: https://mlsanity.com/paper/2205.01562/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 May 2022.

## Built on by (most cited first, 3 of 3)

- [Testing Calibration in Nearly-Linear Time](https://mlsanity.com/paper/2402.13187/) (2024)
- [Robust Sublinear Convergence Rates for Iterative Bregman Projections](https://mlsanity.com/paper/2602.01372/) (2026)
- [Hierarchical Forecast Reconciliation on Networks: A Network Flow Optimization Formulation](https://mlsanity.com/paper/2505.03955/) (2025)

## Abstract (arXiv)

We present a nearly-linear time algorithm for finding a minimum-cost flow in planar graphs with polynomially bounded integer costs and capacities. The previous fastest algorithm for this problem is based on interior point methods (IPMs) and works for general sparse graphs in O(n1.5poly(log n)) time [Daitch-Spielman, STOC'08]. Intuitively, Omega(n1.5) is a natural runtime barrier for IPM-based methods, since they require √n iterations, each routing a possibly-dense electrical flow. To break this barrier, we develop a new implicit representation for flows based on generalized nested-dissection [Lipton-Rose-Tarjan, JSTOR'79] and approximate Schur complements [Kyng-Sachdeva, FOCS'16]. This implicit representation permits us to design a data structure to route an electrical flow with sparse demands in roughly √n update time, resulting in a total running time of O(n·poly(log n)). Our results immediately extend to all families of separable graphs.