Integrality gaps for Sherali-Adams relaxations

We prove strong lower bounds on integrality gaps of Sherali?Adams relaxations for MAX CUT, Vertex Cover, Sparsest Cut and other problems. Our constructions show gaps for Sherali?Adams relaxations that survive n rounds of lift and project. For MAX CUT and Vertex Cover, these show that even n rounds of Sherali?Adams do not yield a better than 2 - approximation. The main combinatorial challenge in constructing these gap examples is the construction of a fractional solution that is far from an integer solution, but yet admits consistent distributions of local solutions for all small subsets of variables. Satisfying this consistency requirement is one of the major hurdles to constructing Sherali?Adams gap examples. We present a modular recipe for achieving this, building on previous work on metrics with a local?global structure. We develop a conceptually simple geometric approach to constructing Sherali?Adams gap examples via constructions of consistent local SDP solutions. This geometric...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Moses Charikar, Konstantin Makarychev, Yury Makarychev
• Keywords: Algorithms, MAX CUT, STOC 2009, Unique Games