Faster and simpler width-independent parallel algorithms for positive semidefinite programming

Abstract

This paper studies the problem of finding a (1+e)-approximate solution to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all scalars are non-negative. At FOCS'11, Jain and Yao gave an NC algorithm that requires O(t 1/e13 log13 m log n) iterations on input n constraint matrices of dimension m-by-m, where each iteration performs at least Ω(mω) work since it involves computing the spectral decomposition. We present a simpler NC parallel algorithm that on input with n constraint matrices, requires O(1/e4 log4 n log(1/e)) iterations, each of which involves only simple matrix operations and computing the trace of the product of a matrix exponential and a positive semidefinite matrix. Further, given a positive SDP in a factorized form, the total work of our algorithm is nearly-linear in the number of non-zero entries in the factorization. Our algorithm can be viewed as a generalization of Young's algorithm and analysis techniques for positive linear programs (Young, FOCS'01) to the semidefinite programming setting.