## Transitive packing (1996)

Venue: | Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science 1084 |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Müller96transitivepacking,

author = {Rudolf Müller and Andreas S. Schulz},

title = {Transitive packing},

booktitle = {Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science 1084},

year = {1996},

pages = {430--444},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper is intended to give a concise understanding of the facial structure of previously separately investigated polyhedra. It introduces the notion of transitive packing and the transitive packing polytope and gives cutting plane proofs for huge classes of valid inequalities of this polytope. We introduce generalized cycle, generalized clique, generalized antihole, generalized antiweb, generalized web, and odd partition inequalities. These classes subsume several known classes of valid inequalities for several of the special cases but give also many new inequalities for several other special cases. For some of the classes we prove as well a lower bound for their Chv'atal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering as well as to balanced and ideal matrices.

### Citations

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1139 | Geometric Algorithms and Combinatorial Optimizations - Grotschel, Lovász, et al. - 1993 |

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362 |
The ellipsoid method and its consequences in combinatorial optimization
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(Show Context)
Citation Context ...edge -- still open, we concentrate on generalized cycle, generalized clique, and odd partition inequalities. In fact, there is not much hope for clique inequalities. Grotschel, Lov'asz, and Schrijver =-=[GLS81]-=- showed that the optimization problem for the clique--constrained node packing polytope is NP--hard. Moreover, Muller [Mul93] proved that the membership problem for the polytope defined by the nonnega... |

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- 1973
(Show Context)
Citation Context ...by a constant independent of n the transitive packing polytope has arbitrarily high rank. It was proved before for the special instances formed by the clique inequalities of the node packing polytope =-=[Chv73]-=- and by the simple k--fence inequalities of the acyclic subdigraph polytope [CCH89]. Notice that in case of the (1; k)--2--partition inequalities of the clique partitioning polytope the assumption of ... |

108 |
On the cut polytope
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(Show Context)
Citation Context ...ve setting. Notice that the inequalities presented above remain valid when we allow for hypergraphs with loops. Then, we cover, for instance, the boolean quadric polytope [Pad89] and the cut polytope =-=[BM86]-=-, as well. It is well--known (see [Edm62]) that every set packing problem maxfc x : Axs1l; xu 2 f0; 1gg where A is a 0=1 matrix can be transformed into an equivalent node packing problem on the inters... |

100 |
On the facial structure of set packing polyhedra
- Padberg
- 1973
(Show Context)
Citation Context ...ualities for the transitive packing polytope P TP (N; H; tr) to the three selected examples. For the node packing polytope we obviously obtain exactly the odd cycle inequalities introduced by Padberg =-=[Pad73]-=-. This is because all edges of the (hyper)graph have size 2 and hence all sets N i have to be singletons. The class of Mobius ladder inequalities is a very prominent class of facet defining inequaliti... |

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34 |
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- 1993
(Show Context)
Citation Context ...ties, introduced by Grotschel and Wakabayashi [GW90]. Also structures that are not cycles in G lead to generalized (k; 2)--cycle inequalities, for instance, the odd wheel inequality of Chopra and Rao =-=[CR93]-=-. Euler, Junger, and Reinelt [EJR87] introduced generalized cycle inequalities for the independence system polytope. Our generalized cycles, if restricted to independence systms, extend those, since t... |

32 |
On cutting-plane proofs in combinatorial optimization. Linear Algebra and its Applications
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(Show Context)
Citation Context ... rank. It was proved before for the special instances formed by the clique inequalities of the node packing polytope [Chv73] and by the simple k--fence inequalities of the acyclic subdigraph polytope =-=[CCH89]-=-. Notice that in case of the (1; k)--2--partition inequalities of the clique partitioning polytope the assumption of Theorem 9 is satisfied, i. e., Theorem 9 applies, for instance, to these inequaliti... |

28 |
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(Show Context)
Citation Context ...een as obtained from P by one step of rounding. If we define P (0) := P and, recursively, P (t+1) := (P (t) ) 0 for all positive integers t, then P I ` P (t) for all nonnegative integers t. Schrijver =-=[Sch80]-=- showed that P 0 is again a polyhedron and that there is a nonnegative integer t such that P (t) = P I . The rank of P is the smallest t such that P (t) = P I . Let a xsfi be a valid inequality for P ... |

24 |
Facets of the clique partitioning polytope
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(Show Context)
Citation Context ... two known classes of inequalities that are even facet defining if G is the complete graph. The first class is formed by the 2--chorded odd cycle inequalities, introduced by Grotschel and Wakabayashi =-=[GW90]-=-. Also structures that are not cycles in G lead to generalized (k; 2)--cycle inequalities, for instance, the odd wheel inequality of Chopra and Rao [CR93]. Euler, Junger, and Reinelt [EJR87] introduce... |

22 |
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- 1985
(Show Context)
Citation Context ...r the independence system with ground set C and circuits H i 1 ;:::;i q only. Euler, Junger, and Reinelt observed also that in case of the acyclic subdigraph polytope the simple k--fence inequalities =-=[GJR85b]-=- are contained in the class of generalized clique inequalities. We can even show that the facet defining k--fence inequalities (not necessarily simple) are contained in the class of generalized (k; 2)... |

22 | Covering, packing and knapsack problems - Padberg - 1979 |

17 | On the set covering polytope: I. all the facets with coefficients - Balas, NG - 1989 |

13 |
A generalization of antiwebs to independence systems and their canonical facets
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- 1989
(Show Context)
Citation Context ...or the independence system polytope: generalized cycle, generalized clique, and generalized antihole inequalities by Euler, Junger, and Reinelt [EJR87] and generalized antiweb inequalities by Laurent =-=[Lau89]-=-. It will turn out that our inequalities are more general even if we restrict to the independence system polytope. Nevertheless, in order to keep the terminology simple we will give them the same name... |

12 |
Facets of the bipartite subgraph polytope
- Barahona, Grötschel, et al.
- 1985
(Show Context)
Citation Context ...asses of valid inequalities for special cases of the independence system polytope. Among them are, just to name a few, the acyclic subdigraph polytope [GJR85b, Jun85], the bipartite subgraph polytope =-=[BGM85]-=-, and the planar subgraph polytope [JM93]. Only recently the thorough study of the facial structure of the independence system polytope in general has begun, see [EJR87, Lau89]. In Sect. 2, we introdu... |

11 |
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- 1989
(Show Context)
Citation Context ...is that any result stated for the independence system polytope can be translated to the set covering polytope, and vice versa. Thus the work of Balas and Ng [BN89a, BN89b], of Cornu'ejols and Sassano =-=[CS89]-=-, Nobili and Sassano [NS89], as well as Sassano [Sas89] on the set covering polytope can be seen as a contribution to the knowledge of the independence system polytope. This implies especially that ou... |

11 | Polyhedral combinatorics and the Acyclic Subdigraph Problem (Heldermann - Jünger - 1985 |

11 | On the partial order polytope of a digraph
- Müller
(Show Context)
Citation Context ...as finding an independent set of maximum weight there are the clique partitioning problem [GW89, GW90, ORS95], the transitive acyclic subdigraph problem [Mul93], the interval graph completion problem =-=[MS95]-=-, and the relatively transitive subgraph problem [KL89, SB]. One of our main purposes is to derive broad classes of valid inequalities for the transitive packing polytope, the convex hull of feasible ... |

10 |
On the facial structure of the set covering polytope
- Sassano
- 1989
(Show Context)
Citation Context ...polytope can be translated to the set covering polytope, and vice versa. Thus the work of Balas and Ng [BN89a, BN89b], of Cornu'ejols and Sassano [CS89], Nobili and Sassano [NS89], as well as Sassano =-=[Sas89]-=- on the set covering polytope can be seen as a contribution to the knowledge of the independence system polytope. This implies especially that our extension of the class of antiweb inequalities for th... |

9 |
Covers and packings in a family of sets
- Edmonds
- 1962
(Show Context)
Citation Context ... presented above remain valid when we allow for hypergraphs with loops. Then, we cover, for instance, the boolean quadric polytope [Pad89] and the cut polytope [BM86], as well. It is well--known (see =-=[Edm62]-=-) that every set packing problem maxfc x : Axs1l; xu 2 f0; 1gg where A is a 0=1 matrix can be transformed into an equivalent node packing problem on the intersection graph of A. Hence transitive packi... |

9 | Trotter Jr. Properties of vertex packing and independence system polyhedra - Nemhauser, E - 1974 |

8 |
Generalizations of cliques, odd cycles and anticycles and their relation to independence system polyhedra
- Euler, Junger, et al.
- 1987
(Show Context)
Citation Context ...alities which we introduce have been presented earlier for the independence system polytope: generalized cycle, generalized clique, and generalized antihole inequalities by Euler, Junger, and Reinelt =-=[EJR87]-=- and generalized antiweb inequalities by Laurent [Lau89]. It will turn out that our inequalities are more general even if we restrict to the independence system polytope. Nevertheless, in order to kee... |

8 | Solving the maximum weight planar subgraph problem by branch and cut
- Jünger, Mutzel
- 1993
(Show Context)
Citation Context ...ses of the independence system polytope. Among them are, just to name a few, the acyclic subdigraph polytope [GJR85b, Jun85], the bipartite subgraph polytope [BGM85], and the planar subgraph polytope =-=[JM93]-=-. Only recently the thorough study of the facial structure of the independence system polytope in general has begun, see [EJR87, Lau89]. In Sect. 2, we introduce in extension of the node packing probl... |

8 |
On the Transitive Acyclic Subdigraph Polytope
- Muller
- 1993
(Show Context)
Citation Context ... Indeed, besides those that can be interpreted as finding an independent set of maximum weight there are the clique partitioning problem [GW89, GW90, ORS95], the transitive acyclic subdigraph problem =-=[Mul93]-=-, the interval graph completion problem [MS95], and the relatively transitive subgraph problem [KL89, SB]. One of our main purposes is to derive broad classes of valid inequalities for the transitive ... |

6 | On the set covering polytope. II. Lifting the facets with coe cients - BALAS, NG - 1989 |

6 |
Facets and lifting procedures for the set covering polytope
- Nobili, Sassano
- 1989
(Show Context)
Citation Context ...or the independence system polytope can be translated to the set covering polytope, and vice versa. Thus the work of Balas and Ng [BN89a, BN89b], of Cornu'ejols and Sassano [CS89], Nobili and Sassano =-=[NS89]-=-, as well as Sassano [Sas89] on the set covering polytope can be seen as a contribution to the knowledge of the independence system polytope. This implies especially that our extension of the class of... |

3 | Acyclic subdigraph and linear orderings: Polytopes, facets and a cutting plane algorithm - Grotschel, Junger, et al. - 1985 |

3 |
A class of facet producing graphs for vertex packing polyhedra
- Jr
- 1975
(Show Context)
Citation Context ...ity is to combine generalized clique inequalities in a manner oriented on the cutting plane proof of generalized cycle inequalities. This leads for the stable set polytope to the antiweb inequalities =-=[Tro75]-=-. For integers 1sssk, a (k; s)--antiweb is a graph with node set A = fu 1 ; u 2 ; : : : ; u k g such that each node u i is adjacent to all other nodes but not to u i+s ; u i+s+1 ; : : : ; u i+k\Gammas... |

2 |
f0; 2 g-Chvatal-Gomory Cuts
- Caprara
- 1996
(Show Context)
Citation Context ...qualities In this section, we introduce another new class of inequalities for the transitive packing polytope. It is an extension of a class of inequalities recently proposed by Caprara and Fischetti =-=[CF95]-=- for the acyclic subdigraph polytope. Let H 1 ; : : : ; H k be a collection of distinct edges of H, and let m(u) := jfi 2 f1; : : : ; kg : u 2 H i gj and n(u) := jfi 2 f1; : : : ; kg : u 2 tr(H i )gj.... |

2 | Polyhedral results for antimatroids - Korte, Lov'asz - 1989 |

2 |
The clique partitioning polytope: Facets
- Oosten, Rutten, et al.
- 1995
(Show Context)
Citation Context ...hull of the incidence vectors of all clique partitionings in G. It was introduced by Grotschel and Wakabayashi [GW89, GW90] and has very recently been considered again by Oosten, Rutten, and Spieksma =-=[ORS95]-=-. To show that it is an instance of the transitive packing polytope we can take as hypergraph the line graph of G. Indeed, we take as the set N of nodes the edges of G, and two nodes are adjacent (for... |

2 |
The boolean quadric polytpe: Some characteristics, facets and relatives
- Padberg
- 1989
(Show Context)
Citation Context ...ebs, and webs in the respective setting. Notice that the inequalities presented above remain valid when we allow for hypergraphs with loops. Then, we cover, for instance, the boolean quadric polytope =-=[Pad89]-=- and the cut polytope [BM86], as well. It is well--known (see [Edm62]) that every set packing problem maxfc x : Axs1l; xu 2 f0; 1gg where A is a 0=1 matrix can be transformed into an equivalent node p... |

2 | On the polyhedral structure of relatively transitive subgraphs - Shallcross, Bland - 1993 |

1 |
Balanced 0; \Sigma1-matrices, bicoloring and total dual integrality
- Conforti, Cornu'ejols
- 1995
(Show Context)
Citation Context ...ver, from the transitive packing polytope we cannot derive generalized set packing polytopessas special instances, neither in the sense of Laurent [Lau89] nor in the sense of Conforti and Cornu'ejols =-=[CC92]-=-. Laurent called convfx 2 f0; 1g n : Axsbg the generalized set packing polytope where A is a 0=1 matrix and b a vector with nonzero integer components. Given a 0= \Sigma 1 matrix A and the vector nA w... |

1 |
0; \Sigma1) ideal matrices
- Nobili, Sassano
- 1998
(Show Context)
Citation Context ...et covering polytope. We call it fractional generalized set covering polytope here and rather Q(A) the generalized set covering polytope. Conforti and Cornu'ejols [CC92] as well as Nobili and Sassano =-=[NS95]-=- try to characterize when the fractional generalized set covering polytope is integral, i. e., when it coincides with the generalized set covering polytope. Our work can be seen as a first contributio... |