Extension Complexity of Stable Set Polytopes of Bipartite Graphs

The extension complexity xc(P ) of a polytope P is the minimum number of facets of a polytope that affinely projects to P . Let G be a bipartite graph with n vertices, m edges, and no isolated vertices. Let STAB(G) be the convex hull of the stable sets of G. Since G is perfect, it is easy to see that n 6 xc(STAB(G)) 6 n+m. We improve both of these bounds. For the upper bound, we show that xc(STAB(G)) is O( n 2 log n ), which is an improvement when G has quadratically many edges. For the lower bound, we prove that xc(STAB(G)) is Ω(n log n) when G is the incidence graph of a finite projective plane. We also provide examples where the obvious upper and lower bounds are both essentially tight.