Solving Linear Programs with Õ ( √ rank ) Linear System Solves

We present an algorithm that given a linear program with n variables, m constraints, and constraint matrix A, computes an -approximate solution in Õ( √ rank(A) log(1/ )) iterations with high probability. Each iteration of our method consists of solving Õ(1) linear systems and additional nearly linear time computation, improving by a factor of Ω̃((m/ rank(A))) over the previous fastest method with this iteration cost due to Renegar (1988) [51].1 Further, we provide a deterministic polynomial time computable Õ(rank(A))-self-concordant barrier function for the polytope, resolving an open question of Nesterov and Nemirovski (1994) [47] on the theory of “universal barriers” for interior point methods. Applying our techniques to the linear program formulation of maximum flow yields an Õ(|E| √ |V | log(U)) time algorithm for solving the maximum flow problem on directed graphs with |E| edges, |V | vertices, and integer capacities of size at most U . This improves upon the previous fastest polynomial running time of O(|E|min{|E|, |V |} log(|V |/|E|) log(U)) achieved by Goldberg and Rao (1998) [18]. In the special case of solving dense directed unit capacity graphs our algorithm improves upon the previous fastest running times ofO(|E|min{|E|, |V |}) achieved by Even and Tarjan (1975) [16] and Karzanov (1973) [22] and of Õ(|E|) achieved more recently by Mądry (2013) [39]. This paper is a journal version of the paper, “Path-Finding Methods for Linear Programming : Solving Linear Programs in Õ( √ rank) Iterations and Faster Algorithms for Maximum Flow” [34] and arXiv submissions [32, 33]. This paper contains several new results beyond these prior submissions. This paper provides the first proof of a Õ(r)-self-concordant barrier for all polytopes {x ∈ R : Ax ≥ b} with r = rank(A) that is polynomial time computable (as opposed to the pseudo-polynomial time computability of the universal barrier of [47]). Further, this paper provides a conceptually simpler weight function than that in our prior works and a unified analysis by establishing new connections between the algorithm, the barrier, and `p Lewis weights [37, 9, 8]. Several components of [34, 32, 33] were not included in this journal version. Techniques, for leveraging this paper to solve linear programs exactly are deferred to [32] and techniques for analyzing the error induced by approximate linear system solves are deferred to [33]. These techniques are fairly standard and general and omitted from this paper for brevity. Further, techniques for reducing the cost of the linear systems found in [32] are also not included and have been improved in a sequence of recent work [35, 8, 2] and techniques solving generalized minimum cost flow as opposed to more restricted minimum cost flow problem considered in this paper are deferred to [33]. ar X iv :1 91 0. 08 03 3v 1 [ cs .D S] 1 7 O ct 2 01 9