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

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


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.

Formats available

You can view the full content in the following formats:

Information & Authors


Published In

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


Published online: 5 January 2022



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options


View PDF

Get Access







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account