## Cones of matrices and set-functions and 0-1 optimization (1991)

Venue: | SIAM JOURNAL ON OPTIMIZATION |

Citations: | 260 - 7 self |

### BibTeX

@ARTICLE{Lovász91conesof,

author = {L. Lovász and A. Schrijver},

title = {Cones of matrices and set-functions and 0-1 optimization},

journal = {SIAM JOURNAL ON OPTIMIZATION},

year = {1991},

volume = {1},

pages = {166--190}

}

### Years of Citing Articles

### OpenURL

### 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. We develop a general method 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, we obtain a sequence of systems of inequalities, such that already the first system 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. We also discuss an extension of the method, which establishes a connection with certain submodular functions and the Möbius function of a lattice.

### Citations

1139 | Geometric Algorithms and Combinatorial Optimization - Grötschel, Lovász, et al. - 1981 |

362 | The ellipsoid method and its consequences in combinatorial optimization - GRÖTSCHEL, LOVÁSZ, et al. - 1981 |

301 |
Maximum matching and a polyhedron with 0, 1-vertices
- Edmonds
- 1965
(Show Context)
Citation Context ...he class of linegraphs: the stable set problem for these graphs is equivalent to the matching problem. In particular, it is polynomial time solvable and Edmonds’ description of the matching polytope (=-=Edmonds 1965-=-) provides a “nice” system of linear inequalities describing the stable set polytope of such graphs. The N-index of line-graphs is unbounded; this 24sfollows e.g. by Corollary 2.8. This also follows f... |

185 | A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems - Sherali, Adams - 1990 |

103 | On certain polytopes associated with graphs - Chvátal - 1975 |

21 | Relaxations of Vertex Packing - Grötschel, Lovasz, et al. - 1986 |

18 | The Steiner problem in graphs - Maculan - 1987 |

11 |
Hadamard determinants, Möbius functions, and the chromatic number of a graph
- Wilf
- 1968
(Show Context)
Citation Context ...iate with it the matrix W f = (wij), where j≥i wij = f(i ∨ j). We also consider the diagonal matrix D f with (D f )ii = f(i). Then it is not difficult to prove the following identity (Lindström 1969, =-=Wilf 1968-=-): 3.1 Lemma. If g is the upper Möbius inverse of f then W f = ZD g Z T . 28sFor more on Möbius functions see Rota (1964), Lovász (1979, Chapter 2), or Stanley (1986, Chapter 3). A function f ∈ IR L w... |

10 |
Determinants on semilattices
- Lindström
- 1969
(Show Context)
Citation Context ... IR L , we associate with it the matrix W f = (wij), where j≥i wij = f(i ∨ j). We also consider the diagonal matrix D f with (D f )ii = f(i). Then it is not difficult to prove the following identity (=-=Lindström 1969-=-, Wilf 1968): 3.1 Lemma. If g is the upper Möbius inverse of f then W f = ZD g Z T . 28sFor more on Möbius functions see Rota (1964), Lovász (1979, Chapter 2), or Stanley (1986, Chapter 3). A function... |

4 | The perfect matchable subgraph polytope of a bipartite graph - Balas, Pulleyblank - 1983 |

4 | Reducing matching to polynomial size linear programming - Barahona - 1993 |

3 | Extended formulations and polyhedral projections - Liu - 1988 |

2 | Coflow polyhedra (preprint - Cameron, Edmonds - 1989 |

2 | The Selberg sieve for a lattice, in: ‘Combinatorial Theory and its - Wilson - 1969 |

1 | and A.R.Mahjoub - Barahona - 1987 |

1 | Gomory (1963), An algorithm for integer solutions to linear programs - E |

1 | On tensor powers of integer programs (preprint) G.-C - Pemantle, Propp, et al. - 1989 |