A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs

Kolla and Tulsiani (2007, 2011) and Arora, Barak and Steurer (2010) introduced the technique of subspace enumeration, which gives approximation algorithms for graph problems such as unique games and small set expansion; the running time of such algorithms is exponential in the threshold rank of the graph. Guruswami and Sinop (2011, 2012) and Barak, Raghavendra, and Steurer (2011) developed an alternative approach to the design of approximation algorithms for graphs of bounded threshold rank based on semidefinite programming relaxations obtained by using sum-of-squares hierarchy (2000, 2001) and on novel rounding techniques. These algorithms are faster than the ones based on subspace enumeration and work on a broad class of problems. An extended abstract of this paper appeared in the proceedings of the 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2013) [15]. ∗Department of Computer Science and Engineering, University of Washington. †Department of Electrical Engineering and Computer Sciences U.C. Berkeley. This material is based upon work supported by the National Science Foundation under grant No. CCF 1017403. ACM Classification: F.2.2, G.2.2, G.1.6 AMS Classification: 68W25, 68Q25, 68R10