2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS) (2014)

Philadelphia, PA, USA

Oct. 18, 2014 to Oct. 21, 2014

ISSN: 0272-5428

ISBN: 978-1-4799-6517-5

pp: 424-433

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2014.52

ABSTRACT

In this paper, we present a new algorithm for solving linear programsthat requires only Õ(&#x221A;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 &#x03a9;(m/rank(A))1/4 for methods with polynomial time computable iterations and by &#x03a9;((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| &#x221A;(|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 M&#x0105;dry.

INDEX TERMS

Linear systems, Standards, Linear programming, Convergence, Approximation methods, Polynomials, Laplace equations

CITATION

Y. T. Lee and A. Sidford, "Path Finding Methods for Linear Programming: Solving Linear Programs in Õ(vrank) Iterations and Faster Algorithms for Maximum Flow,"

*2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS)*, Philadelphia, PA, USA, 2014, pp. 424-433.

doi:10.1109/FOCS.2014.52

CITATIONS