Solving Linear Programs in the Current Matrix Multiplication Time

Michael B. Cohen, Yin Tat Lee, Zhao Song

Abstract: This article shows how to solve linear programs of the form minAx=b,x≥ 0 c⊤ x with n variables in time O*((nω+n2.5−α/2+n2+1/6) log (n/δ)), where ω is the exponent of matrix multiplication, α is the dual exponent of matrix multiplication, and δ is the relative accuracy. For the current value of ω δ 2.37 and α δ 0.31, our algorithm takes O*(nω log (n/δ)) time. When ω = 2, our algorithm takes O*(n2+1/6 log (n/δ)) time. Our algorithm utilizes several new concepts that we believe may be of independent interest: • We define a stochastic central path method. • We show how to maintain a projection matrix √ WA⊤ (AWA⊤)−1A√ W in sub-quadratic time under \ell2 multiplicative changes in the diagonal matrix W.