A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Algebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.
"Minimal Resolutions of Lattice Ideals and Integer Linear Programming." Rev. Mat. Iberoamericana 19 (2) 287 - 306, September, 2003.