Integrality gaps for Sherali-Adams relaxations

Moses Charikar, Konstantin Makarychev, Yury Makarychev
2009 Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09  
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 δ rounds of lift and project. For MAX CUT and Vertex Cover, these show that even n δ rounds of Sherali-Adams do not yield a better than 2 − ε approximation. The main combinatorial challenge in constructing these gap examples is the construction of a fractional solution that is far from
more » ... 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 approach is surprisingly versatile. We construct Sherali-Adams gap examples for Unique Games based on our construction for MAX CUT together with a parallel repetition like procedure. This in turn allows us to obtain Sherali-Adams gap examples for any problem that has a Unique Games based hardness result (with some additional conditions on the reduction from Unique Games). Using this, we construct 2 − ε gap examples for Maximum Acyclic Subgraph that rules out any family of linear constraints with support at most n δ .
doi:10.1145/1536414.1536455 dblp:conf/stoc/CharikarMM09 fatcat:dkanhr3ge5hvpcuhdulgfcmrrm