A Parallel Approximation Algorithm for Positive Semidefinite Programming

TL;DR

This paper introduces a parallel approximation algorithm for positive semidefinite programs that efficiently computes near-optimal solutions using polylogarithmic time and polynomial processors, generalizing previous linear program results.

Contribution

It presents the first parallel approximation algorithm for positive semidefinite programs with provable guarantees and polylogarithmic runtime, extending prior work on linear programs.

Findings

Runs in poly(1/eps) * polylog(N) time with polynomial processors

Achieves a (1 + eps)-approximate solution to the PSDP

Generalizes previous linear program approximation algorithms

Abstract

Positive semidefinite programs are an important subclass of semidefinite programs in which all matrices involved in the specification of the problem are positive semidefinite and all scalars involved are non-negative. We present a parallel algorithm, which given an instance of a positive semidefinite program of size N and an approximation factor eps > 0, runs in (parallel) time poly(1/eps) \cdot polylog(N), using poly(N) processors, and outputs a value which is within multiplicative factor of (1 + eps) to the optimal. Our result generalizes analogous result of Luby and Nisan [1993] for positive linear programs and our algorithm is inspired by their algorithm.