A Combinatorial, Primal-Dual Approach to Semidefinite Programs

Abstract

Semidefinite programs (SDPs) have been used in many recent approximation algorithms. We develop a general primal-dual approach to solve SDPs using a generalization of the well-known multiplicative weights update rule to symmetric matrices. For a number of problems, such as S parsest C ut and B alanced S eparator in undirected and directed weighted graphs, M in U n C ut and M in 2CNF D eletion , this yields combinatorial approximation algorithms that are significantly more efficient than interior point methods. The design of our primal-dual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones. Our ideas have proved useful in quantum computing, especially the recent result of Jain et al. [2011] that QIP = PSPACE.