Note on pseudolattices, lattices and submodular linear programs

A pseudolattice LL is a poset with lattice-type binary operations. Given a submodular function r:L→Rr:L→R and a modular representation of the pseudolattice as a family of subsets of a set UU with certain compatibility properties, we demonstrate that the corresponding unrestricted linear program relative to the representing set family can be solved by a greedy algorithm. This complements the Monge algorithm of Dietrich and Hoffman for the associated dual linear program. We furthermore show that our Monge and greedy algorithms are generally optimal for nonnegative submodular linear programs and their duals (relative to LL). Finally, we show that LL actually is a distributive lattice with the same supremum operation. Using Birkhoff’s representation theorem for distributive lattices, we construct a suitable weight function on PP that allows us to reduce the problems to generalized polymatroids.