Nested Dissection Meets IPMs: Planar Min-Cost Flow in Nearly-Linear Time
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Author(s)
Dong, Sally
Gao, Yu
Goranci, Gramoz
Lee, Yin Tat
Sachdeva, Sushant
Peng, Richard
Ye, Guanghao
Date Issued
July 26, 2025
Journal
Journal of the ACM
Publisher
ACM
Citation
Sally Dong, Yu Gao, Gramoz Goranci, Yin Tat Lee, Sushant Sachdeva, Richard Peng, and Guanghao Ye. 2025. Nested Dissection Meets IPMs: Planar Min-Cost Flow in Nearly-Linear Time. J. ACM 72, 4, Article 27 (August 2025), 75 pages.
Version
Final published version
Abstract
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(n1.5 polylog n)) time [Daitch-Spielman, STOC'. Intuitively, ?(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, SINUM'79] and approximate Schur complements [Kyng-Sachdeva, FOCS'. 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 runtime of O(n polylog n). Our results immediately extend to all families of separable graphs.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
http://dx.doi.org/10.1145/3744639
