Complexity of linear relaxations in integer programming.

For a set X of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with  X is called the relaxation complexity  rc ( X ) . This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of  X without using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding rc ( X ) and its variant rc Q ( X ) , restricting the descriptions of  X to rational polyhedra. As our main results we show that rc ( X ) = rc Q ( X ) when: (a) X is at most four-dimensional, (b) X represents every residue class in ( Z / 2 Z ) d , (c) the convex hull of  X contains an interior integer point, or (d) the lattice-width of  X is above a certain threshold. Additionally, rc ( X ) can be algorithmically computed when  X is at most three-dimensional, or X satisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on rc ( X ) in terms of the dimension of  X .