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 capacities bounded by $${{\,\textrm{poly}\,}}n$$ . In this paper, we show how to find an exact solution minimum cost flow problem in $$n^{1/2+o(1)}(\sqrt{n}+D)$$ rounds, improving state of art algorithm running time $$m^{3/7+o(1)}(\sqrt{n}D^{1/4}+D)$$ [16], which only holds for special case unit capacity graphs. For certain graphs, achieve even better results. particular, planar expander $$n^{o(1)}$$ -genus -treewidth excluded-minor graphs our takes $$n^{1/2+o(1)}D$$ rounds. obtain result combining recent results Laplacian solvers [3, 16] implementation LP solver Lee Sidford [29], finally that can round approximate solution. Our solves linear programs, generalize flow, up additive error $$\epsilon $$ $$n^{1/2+o(1)}(\sqrt{n}+D)\log ^3 (1/\epsilon )$$