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 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, $Ω(n^{1.5})$ 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.