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

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Sally Dong, Yu Gao, Gramoz Goranci, Yin Tat Lee, Sushant Sachdeva, Richard Peng, Guanghao Ye

Published: 2025, Last Modified: 18 Nov 2025J. ACM 2025EveryoneRevisionsBibTeXCC BY-SA 4.0
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 ⋅ poly (log n)) time [Daitch-Spielman, STOC’08].Intuitively, Ω (n1.5) is a natural runtime barrier for IPM-based methods, since they require \(\sqrt {n}\) 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’16]. This implicit representation permits us to design a data structure to route an electrical flow with sparse demands in roughly \(\sqrt {n}\) update time, resulting in a total runtime of O(n ⋅ poly (log n)).Our results immediately extend to all families of separable graphs.