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

TL;DR

This paper introduces a nearly-linear time algorithm for minimum-cost flow in planar graphs by combining nested dissection and interior point methods, surpassing previous runtime barriers for such problems.

Contribution

The authors develop a new implicit flow representation using nested dissection and approximate Schur complements, enabling faster electrical flow routing and breaking the traditional $O(n^{1.5})$ barrier.

Findings

Achieved nearly-linear time complexity for planar min-cost flow.

Extended results to all separable graph families.

Provided a new data structure for sparse electrical flow routing.

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 $O(n^{1.5}\text{poly}(\log n))$ time [Daitch-Spielman, STOC'08]. Intuitively, $\Omega(n^{1.5})$ 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, 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 $\sqrt{n}$ update time, resulting in a total running time of $O(n\cdot\text{poly}(\log n))$. Our results immediately extend to all families of separable graphs.