# Approximate constraint satisfaction requires sub-exponential size linear programs

## Approximate constraint satisfaction requires sub-exponential size linear programs - Pravesh Kothari

Pravesh Kothari
Member, School of Mathematics
December 6, 2016
We show that for constraint satisfaction problems (CSPs), sub-exponential size linear programming relaxations are as powerful as $n^{\Omega(1)}$-rounds of the Sherali-Adams linear programming hierarchy. As a corollary, we obtain sub-exponential 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 $\sim n^{(\log n)}$ cannot beat random guessing for any CSP [Chan-Lee-Raghavendra-Steurer 2013]. 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 of independent interest in communication complexity. Based on joint work with Raghu Meka and Prasad Raghavendra.