Ordered Sets and Linear Programming

Abstract

Our themes are: the use of linear programming ideas to prove theorems about ordered sets, and the use of ideas from ordered sets to prove theorems in linear programming.

In the first group, we will discuss Dilworth’s theorem (for which we will present two linear programming demonstrations, one using the notion of a totally unimodular matrix [5], the other using nothing [22]) and the generalizations of it by Greene and Kleitman ([16],[17]), which we generalize still further [24], An amusing sidelight is that their results on k- and k+l-saturation are derived in the linear programming context as properties of parametric linear programs, and are not needed (as they are in G-K) to prove the main theorems.

It is proper also to think of the max flow min cut theorem as a theorem on ordered sets, if one returns to the original Ford-Fulkerson paper[[11]. We shall present a generalization [20] which emphasizes that connection.

Analogous to the interplay between chains and antichains in Dilworth’s theorem is the relation between paths and cuts in the max flow min cut theorem. This leads us to consider the cut- packing theorem, for which we also offer a linear programming proof.

We turn to the second group of topics. Here our principal tool is the concept of lattice polyhedron [25], whose defining rows are indexed by elements of a lattice. These polyhedra have the property that all their vertices are integral and also that certain “dual” problems have, among their optimal vertices, atleast one which is integral. We show that many theorems asserting integrality of linear programming problems are subsumed by this idea, including some of the theorems discussed in the first group, and we give details about the “most general” lattice polyhedra [19].

Finally, we use the concept to give a characterization of the lattice of cuts of the paths considered in the max flow min cut theorem [21].