Fundamental domains for integer programs with symmetries

Abstract

We define a fundamental domain of a linear programming relaxation of a combinatorial integer program which is symmetric under a group action. We then provide a construction for the polytope of a fundamental domain defined by the maximization of a linear function. The computation of this fundamental domain is at worst polynomial in the size of the group. However, for the special case of the symmetric group, whose size is exponential in the size of the integer program, we show how to compute a separating hyperplane in polynomial time in the size of the integer program. Fundamental domains may provide a straightforward way to reduce the computation difficulties that often arise in integer programs with symmetries. Our construction is closely related to the constructions of orbitopes by Kaibel and Pfetch, but are simpler and more general, at a cost of creating new non-integral extreme points. © Springer-Verlag Berlin Heidelberg 2007.