From Weak to Strong Linear Programming Gaps for All Constraint Satisfaction Problems.

We study the approximability of constraint satisfaction problems (CSPs) by linear programming (LP) relaxations. We show that for every CSP, the approximation obtained by a basic LP relaxation is at least as strong as the approximation obtained using relaxations given by c . log n/ log log n levels of the Sherali-Adams hierarchy (for some constant c > 0) on instances of size n. It was proved by Chan et al. [FOGS 2013] (and recently strengthened by Kothari et al. [STOC 2017]) that for CSPs, any polynomial-size LP extended formulation is at most as strong as the relaxation obtained by a constant number of levels of the Sherali-Adams hierarchy (where the number of levels depend on the exponent of the polynomial in the size bound). Combining this with our result also implies that any polynomial-size LP extended formulation is at most as strong as the basic LP, which can be thought of as the base level of the Sherali-Adams hierarchy. This essentially gives a dichotomy result for approximation of CSPs by polynomial-size LP extended formulations. Using our techniques, we also simplify and strengthen the result by Khot et al. [STOC 2014] on (strong) approximation resistance for LPs. They provided a necessary and sufficient condition under which o(log log n) levels of the Sherali-Adams hierarchy cannot achieve an approximation better than a random assignment. We simplify their proof and strengthen the bound to o(log n/loglog n) levels.