LS+ Lower Bounds from Pairwise Independence

Madhur Tulsiani, Pratik Worah

Abstract: We consider the complexity of LS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sub> refutations of unsatisfiable instances of Constraint Satisfaction Problems (k-CSPs) when the underlying predicate supports a pairwise independent distribution on its satisfying assignments. This is the most general condition on the predicates under which the corresponding MAX k-CSP problem is known to be approximation resistant. We show that for random instances of such k-CSPs on n variables, even after Ω(n) rounds of the LS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sub> hierarchy, the integrality gap remains equal to the approximation ratio achieved by a random assignment. In particular, this also shows that LS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sub> refutations for such instances require rank Ω(n). We also show the stronger result that refutations for such instances in the static LS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sub> proof system requires size exp(Ω(n)).