Symmetries of linear programs

Abstract

The symmetric groups Sn and the cyclic groups Cn essentially are the only examples for symmetry groups of linear or integer programs that have been discussed in the literature, see e.g. [5] and [6]. In [4], Bodi, Herr, and Joswig developed some ideas to tackle linear and integer programs with arbitrary groups of symmetries. However, the question remained whether or not there are linear (integer) programs with groups of symmetries other than Sn and Cn. Indeed, we show in this short note that every finite permutation group is the full symmetry group of a suitable linear or integer program. Some of our constructions are based on graph theory.