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 (O) over tilde (m + n(1.5)) time. This improves upon the previous best runtime of (O) over tilde (m root n) [Lee-Sidford'14] and, in the special case of unit-capacity maximum flow, improves upon the previous best runtimes of m(4/3+o(1)) [Liu-Sidford'20, Kathuria'20] and (O) over tilde (m root n) [Lee-Sidford'14] for sufficiently dense graphs. In the case of l(1)-regression in a matrix with n-columns and m-rows we obtain a randomized method which computes an epsilon-approximate solution in (O) over tilde (mn + n(2.5)) time. This yields a randomized method which computes an epsilon-optimal policy of a discounted Markov Decision Process with S states and, A actions per state in time (O) over tilde (S(2)A + S-2.5). These methods improve upon the previous best runtimes of methods which depend polylogarithmically on problem parameters, which were (O) over tilde (mn(1.5) ) [Lee-Sidford'15] and (O) over tilde (S-2.5 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.