APPROXIMATING RECTANGLES BY JUNTAS AND WEAKLY EXPONENTIAL LOWER BOUNDS FOR LP RELAXATIONS OF CSPS

We show that for constraint satisfaction problems (CSPs), weakly exponential size linear programming relaxations are as powerful as the explicit linear program described by n^Ω(1)rounds of the Sherali–Adams linear programming hierarchy. Combining with the known lower bounds on the Sherali–Adams hierarchy, we obtain subexponential size lower bounds for linear programming relaxations that beat random guessing for many CSPs such as MAX-CUT and MAX-3SAT. This is a nearly exponential improvement over previous results; previously, it was only known that linear programs of size n^o(log n) cannot beat random guessing for such CSP Chan et al. [FOCS 2013, IEEE, Piscataway, NJ, 2013, pp. 350–359]. Our bounds are obtained by exploiting and extending the recent progress in communication complexity for “lifting” query lower bounds to communication problems. The main ingredient in our results is a new structural result on “high-entropy rectangles” that may be of independent interest in communication complexity.