Tropical lower bound for extended formulations. II. Deficiency graphs of matrices

Abstract

Deficiency graphs arise in the problem of decomposing a tropical vector into a sum of points of a given tropical variety. We give an application of this concept to the theory of extended formulations of convex polytopes, and we show that the chromatic number of the deficiency graph of a special tropical matrix is a lower bound for the extension complexity of the corresponding convex polytope. We compare our new lower bound for extended formulations with existing estimates and make several conjectures on the relations between deficiency graphs, extended formulations, and rank functions of tropical matrices.