Cones of Matrices and Set-Functions and 0–1 Optimization

TLDR

It has been recognized that representing a polyhedron as the projection of a higher‑dimensional, simpler polyhedron is a powerful tool in polyhedral combinatorics. The authors develop a general method to construct higher‑dimensional polyhedra whose projection approximates the convex hull of 0–1 solutions of a system of linear inequalities. The method generates sequences of inequalities that, for the vertex packing polytope, include clique, odd‑hole, odd‑antihole, wheel, and orthogonality constraints, and extends to connect with submodular functions and the Möbius function of a lattice. The approximations enable polynomial‑time optimization of any linear objective, and for perfect and many other graphs the first system yields the exact vertex packing polytope, while for classes such as t‑perfect graphs the stable set polytope is the projection of a polytope with polynomially many facets.

Abstract

It has been recognized recently that to represent a polyhedron as the projection of a higher-dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. A general method is developed to construct higher-dimensional polyhedra (or, in some cases, convex sets) whose projection approximates the convex hull of 0–1 valued solutions of a system of linear inequalities. An important feature of these approximations is that one can optimize any linear objective function over them in polynomial time. In the special case of the vertex packing polytope, a sequence of systems of inequalities is obtained such that the first system already includes clique, odd hole, odd antihole, wheel, and orthogonality constraints. In particular, for perfect (and many other) graphs, this first system gives the vertex packing polytope. For various classes of graphs, including t-perfect graphs, it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets. An extension of the method is also discussed which establishes a connection with certain submodular functions and the Möbius function of a lattice.