Approximate constraint satisfaction requires sub-exponential size linear programs

Computer Science/Discrete Mathematics Seminar II
Topic:Approximate constraint satisfaction requires sub-exponential size linear programs
Speaker:Pravesh Kothari
Affiliation:Member, School of Mathematics
Date:Tuesday, December 6
Time/Room:10:30am - 12:30pm/S-101
Video Link:https://video.ias.edu/csdm/2016/1206-PraveshKothari

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.