Path Finding Methods for Linear Programming: Solving Linear Programs in Õ(vrank) Iterations and Faster Algorithms for Maximum Flow

In this paper, we present a new algorithm for solving linear programsthat requires only Õ(√rank(A)L) iterationswhere A is the constraint matrix of a linear program with mconstraints, n variables, and bit complexity L. Each iterationof our method consists of solving Õ(1) linear systems andadditional nearly linear time computation. Our method improves uponthe previous best iteration bounds by factor of Ω(m/rank(A))1/4 for methods with polynomial time computable iterations and by Ω((m/rank(A))1/2) for methods which solve at most Õ(1) linear systems ineach iteration each achieved over 20 years ago.Applying our techniques to the linear program formulation of maximumflow yields an Õ(|E| √(|V|) log2(U)) time algorithm forsolving the maximum flow problem on directed graphs with |E| edges,|V| vertices, and capacity ratio U. This improves upon the previousfastest running time of O(|E| min(|E|(1/2), |V|(2/3)) log(|V|2/|E|) log(U))achieved over 15 years ago by Goldberg and Rao and improves upon theprevious best running times for solving dense directed unit capacitygraphs of O(|E| min(|E|(1/2), |V|(2/3))) achieved by Even and Tarjanover 35 years ago and a running time of Õ(|E|(10/7)) achievedrecently by Mądry.