CS 2429 - Foundations of Communication Complexity

Linear programming is a very powerful tool for attacking hard combinatorial optimization problems. Methods such as the ellipsoid algorithm have shown that linear programming is solvable in polynomial time. Linear programming also plays a central role in the design of approximation algorithms. In fact, it is known that linear programming is P-complete, and this implies that if NP = P then for every problem in NP , given an instance, it is possible (in polytime) to solve it via a polynomial-sized LP. A large class of linear programs were identified by Yannakakis, and referred to as extended formulations. We emphasize that extended formulations of LPs to not capture all LPs for solving a given NP-hard problems, but nonetheless, they capture a large and useful family of LPs. We will define extended formulations, and then prove that lower bounds on extended formulations follow from communication complexity lower bounds.