Minimum cost flows, MDPs, and ℓ1-regression in nearly linear time for dense instances

TL;DR: In this paper, a randomized algorithm with improved runtimes of O(m + n1.5) was proposed to solve the minimum cost flow problem on n-vertex m-edge graphs with integer polynomially bounded costs and capacities.

Abstract: 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 O(m + n1.5) time. This improves upon the previous best runtime of O(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 O(m √n) [Lee-Sidford’14] for sufficiently dense graphs. In the case of l1-regression in a matrix with n-columns and m-rows we obtain a randomized method which computes an є-approximate solution in O(mn + n2.5) time. This yields a randomized method which computes an є-optimal policy of a discounted Markov Decision Process with S states and, A actions per state in time O(S2 A + S2.5). These methods improve upon the previous best runtimes of methods which depend polylogarithmically on problem parameters, which were O(mn1.5) [Lee-Sidford’15] and O(S2.5 A) [Lee-Sidford’14, Sidford-Wang-Wu-Ye’18] respectively. To obtain this result we introduce two new algorithmic tools of possible independent interest. First, we design a new general interior point method for solving linear programs with two sided constraints which combines techniques from [Lee-Song-Zhang’19, Brand et al.’20] to obtain a robust stochastic method with iteration count nearly the square root of the smaller dimension. Second, to implement this method we provide dynamic data structures for efficiently maintaining approximations to variants of Lewis-weights, a fundamental importance measure for matrices which generalize leverage scores and effective resistances.