Summary: A Combinatorial, Primal-Dual approach to Semidefinite Programs
Computer Science Department, Princeton University
35 Olden Street, Princeton, NJ 08540
Semidefinite programs (SDP) 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 Sparsest Cut and
Balanced Separator in undirected and directed weighted graphs, and the Min UnCut problem, 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.
Semidefinite programming (SDP) has proved useful in design of approximation algorithms for NP-hard
problems, and often (as in case of MaxCut, Sparsest Cut, Min UnCut, Min 2CNF Deletion, etc.)
yields better approximation ratios than known LP-based methods.
But in several ways, our understanding of SDPs seriously lags our understanding of LPs. One is running
time: though LP and SDP are syntactically similar when viewed as subcases of cone optimization and can