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Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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

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 was based on interior point methods (IPMs) and worked 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 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 update time, resulting in a total running time of O(n · poly(log n)).
Our results immediately extend to all families of separable graphs.

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cover image Proceedings
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
Pages: 124 - 153
Editors: Joseph (Seffi) Naor, Technion Israel Institute of Technology, Israel and Niv Buchbinder, Tel Aviv University, Israel
ISBN (Online): 978-1-61197-707-3

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Published online: 5 January 2022

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