Fast SDP Algorithms for Constraint Satisfaction Problems

The class of constraint satisfactions problems (CSPs) captures many fundamental combinatorial optimization problems such as Max Cut, Max q-Cut, Unique Games, and Max k-Sat. Recently, Raghavendra (STOC`08) identified a simple semidefinite programming relaxation that gives the best possible approximation for any CSP, assuming the Unique Games Conjecture. Raghavendra and Steurer (FOCS`09) showed that, independent of the truth of the Unique Games Conjecture, the integrality gap of this relaxation cannot be improved even by adding a large class of valid inequalities. We present an algorithm that finds an approximately optimal solution to this relaxation in near-linear time. Combining this algorithm with a rounding scheme of Raghavendra and Steurer (FOCS`09) leads to an approximation algorithm for any CSP that runs in near-linear time and has an approximation guarantee that matches the integrality gap, which is optimal assuming the Unique Games Conjecture.

• Type: Conference
• Year: 2010
• Where: SODA
• Authors: David Steurer
• Keywords: Approximation Algorithm, Discrete Algorithms, Semidefinite Programming Relaxation, SODA 2010, Unique Games Conjecture