WAOA 2016 Invited Talks

Combinatorial problems like TSP optimize a linear function over some polytope P. If we can obtain P as a projection from a larger-dimensional polytope with a small number of facets, then we get a small linear program for the optimization problem; if we obtain P as a projection from a small spectrahedron, then we get a small semidefinite program. The area of extension complexity studies the minimum sizes of such LPs and SDPs. In the 1980s Yannakakis was the first to do this, proving exponential lower bounds on the size of symmetric LPs for the TSP and matching polytopes. In 2012, Fiorini et al. proved exponential lower bounds on the size of all (possibly non-symmetric) LPs for TSP. This was followed by many new results for LPs and SDPs, for exact optimization as well as for approximation. We will survey this recent line of work.