Minimum Cost Flow in the CONGEST Model

We consider the CONGEST model on a network with $n$ nodes, $m$ edges, diameter $D$, and integer costs and capacities bounded by $\text{poly} n$. In this paper, we show how to find an exact solution to the minimum cost flow problem in $n^{1/2+o(1)}(\sqrt{n}+D)$ rounds, improving the state of the art algorithm with running time $m^{3/7+o(1)}(\sqrt nD^{1/4}+D)$ [Forster et al. FOCS 2021], which only holds for the special case of unit capacity graphs. For certain graphs, we achieve even better results. In particular, for planar graphs, expander graphs, $n^{o(1)}$-genus graphs, $n^{o(1)}$-treewidth graphs, and excluded-minor graphs our algorithm takes $n^{1/2+o(1)}D$ rounds. We obtain this result by combining recent results on Laplacian solvers in the CONGEST model [Forster et al. FOCS 2021, Anagnostides et al. DISC 2022] with a CONGEST implementation of the LP solver of Lee and Sidford [FOCS 2014], and finally show that we can round the approximate solution to an exact solution. Our algorithm solves certain linear programs, that generalize minimum cost flow, up to additive error $ε$ in $n^{1/2+o(1)}(\sqrt{n}+D)\log^3 (1/ε)$ rounds.