Cut Lattices, Quasi h-Distances and Hilbert Bases

Abstract

We consider in this chapter several additional questions related to the notion of hypercube embedding. A possible way of relaxing this notion is to look for integer combinations rather than nonnegative integer combinations of cut semimetrics. In other words, one considers the lattice ℒ _n generated by all cut semimetrics on V _n . We recall in Section 25.1 the characterization of ℒ _n . This is an easy result; namely, ℒ _n consists of the integer distances satisfying the parity condition. We also present the characterization of some sublattices of ℒ _n , namely, of the sublattice generated by all even T-cut semimetrics and of the sublattice generated by all k-uniform cut semimetrics.