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

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+n 1.5 ) time. This improves upon the previous best runtime of O(m √ n) (Lee-Sidford 2014) and, in the special case of unit-capacity maximum flow, improves upon the previous best runtimes of m 4/3+o(1) (Liu-Sidford 2020, Kathuria 2020) and O(m √ n) (Lee-Sidford 2014) for sufficiently dense graphs. For ℓ 1 -regression in a matrix with n-columns and m-rows we obtain a randomized method which computes an ǫ-approximate solution in O(mn + n 2.5 ) time. This yields a randomized method which computes an ǫ-optimal policy of a discounted Markov Decision Process with S states and A actions per state in time O(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(mn 1.5 ) (Lee-Sidford 2015) and O(S 2.5 A) (Lee-Sidford 2014, Sidford-Wang-Wu-Ye 2018).To obtain this result we introduce two new algorithmic tools of 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 2019 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.

Untitled

“…Subsequent work further improved the above bound. First, by employing the recent O(m + n 1.5 )-time Max-Flow algorithm [15], our framework immediately gives a better running time O(min{m 3/2 n 1/6 , mn 3/4 + m 3/2 }). Second, new algorithms were developed for simple graphs: it has started with [6], and has culminated in running time n 2+o (1) [5,37], or a slightly faster O(n 2 ) conditionally on a linear-time Max-Flow algorithm [48], and an even better bound for sufficiently sparse graphs min{m 7/6 n 1/2+o (1) , mn 1/2+o(1) + m 1/2 n 5/4+o (1) } [37].…”

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Global Connectivity Problems

“…On one hand we can apply any max flow algorithm in EC(m r , n r ) time. As remarked above we have EC(m r , n r ) ≤ Õ m r + n 1.5 r by [6]. The second approach is to apply blocking flows with the following additional observations.…”

“…Let k = |R|. We apply Theorem 3.1 and give concrete upper bounds using known upper bounds for [36] and EC(m, n) = Õ m + n 3/2 by [6]. Let α > 0 be a parameter to be determined.…”

“…]) plug directly into Theorem 3.2 to give running times of the form Õ(EC(m, n)) to compute the global element connectivity. A recent breakthrough work by Brand et al[6] has obtained a running time of EC(m, n) = Õ m + n 1.5 for polynomially bounded and integral capacities. However, Theorem 3.2 does not directly benefit from this running time because the vertices are not partitioned across subproblems.…”

Dynamic Least-Squares Regression

“…For general matrices A ∈ R n×d , using iterative uniform row sampling and JL sketch, the leverage scores can be approximately computed in O(nnz(A) + d ω ) time [16]. Dynamic maintenance of leverage scores has been studied by [8,7,6] in their specific applications of optimization problems.…”