Minimum cost flows, MDPs, and ℓ ₁ -regression in nearly linear time for dense instances

In this paper we provide new randomized algorithms with improved runtimes for solving linear programs with two-sided constraints. In the special case of the minimum cost flow problem on n-vertex m-edge graphs with integer polynomially-bounded costs and capacities we obtain a randomized method which solves the problem in Õ(m + n1.5) time. This improves upon the previous best runtime of Õ(m √n) [Lee-Sidford’14] and, in the special case of unit-capacity maximum flow, improves upon the previous best runtimes of m4/3 + o(1) [Liu-Sidford’20, Kathuria’20] and Õ(m √n) [Lee-Sidford’14] for sufficiently dense graphs.