Integrality Gaps for Sherali-Adams Relaxations

Yury Makarychev
Microsoft Research New England

We prove strong lower bounds on integrality gaps of Sherali--Adams relaxations for MAX CUT, Vertex Cover, Maximum Acyclic Subgraph, Sparsest Cut and other problems. Our constructions show gaps for Sherali--Adams relaxations that survive n^\delta rounds of lift and project. For MAX CUT, Vertex Cover, and Maximum Acyclic Subgraph, these show that even n^\delta rounds of Sherali--Adams do not yield a better than 2-epsilon 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 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).

Joint work with Moses Charikar and Konstantin Makarychev

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