An extension of disjunctive programming and its impact for compact tree formulations

TL;DR

This paper generalizes disjunctive programming to create compact linear formulations for complex combinatorial problems like Steiner trees and Gomory-Hu trees, enhancing optimization modeling techniques.

Contribution

It extends the polyhedral structure in disjunctive programming, enabling polynomial-size formulations for several combinatorial optimization problems.

Findings

Derived polynomial size linear programming formulations for Steiner trees and Gomory-Hu trees.

Extended the theoretical framework of disjunctive programming with a new polyhedral coupling approach.

Connected the new model with polyhedral branching systems, adding an algorithmic perspective.

Abstract

In the 1970's, Balas introduced the concept of disjunctive programming, which is optimization over unions of polyhedra. One main result of his theory is that, given linear descriptions for each of the polyhedra to be taken in the union, one can easily derive an extended formulation of the convex hull of the union of these polyhedra. In this paper, we give a generalization of this result by extending the polyhedral structure of the variables coupling the polyhedra taken in the union. Using this generalized concept, we derive polynomial size linear programming formulations (compact formulations) for a well-known spanning tree approximation of Steiner trees, for Gomory-Hu trees, and, as a consequence, of the minimum $T$-cut problem (but not for the associated $T$-cut polyhedron). Recently, Kaibel and Loos (2010) introduced a more involved framework called {\em polyhedral branching systems} to derive extended formulations. The most parts of our model can be expressed in terms of their framework. The value of our model can be seen in the fact that it completes their framework by an interesting algorithmic aspect.