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

Dong, S., Gao, Y., Goranci, G., Lee, Y. T., Peng, R., Sachdeva, S. and Ye, G. (2022) Nested Dissection Meets IPMs: Planar Min-Cost Flow in Nearly-Linear Time. In: 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 9-12 Jan 2022, pp. 124-153. ISBN 9781611977073 (doi: 10.1137/1.9781611977073.7)

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.

• Item Type: Conference Proceedings
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Goranci, Dr Gramoz
• Authors: Dong, S., Gao, Y., Goranci, G., Lee, Y. T., Peng, R., Sachdeva, S., and Ye, G.
• College/School: College of Science and Engineering > School of Computing Science
• ISBN: 9781611977073
• Copyright Holders: Copyright © 2022 by SIAM
• Publisher Policy: Reproduced with the permission of the publisher

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