## Transitive packing: A unifying concept in combinatorial optimization (2002)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Müller02transitivepacking:,

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

title = {Transitive packing: A unifying concept in combinatorial optimization},

year = {2002}

}

### OpenURL

### Abstract

This paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately. It introduces the notion of transitive packing and the transitive packing polytope. Polytopes that turn out to be special cases of the transitive packing polytope include the node packing, acyclic subdigraph, bipartite subgraph, planar subgraph, clique partitioning, partition, transitive acyclic subdigraph, interval order, and relatively transitive subgraph polytopes. We give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope, thereby introducing generalized cycle, generalized clique, generalized antihole, generalized antiweb, and odd partition inequalities. On the one hand, these classes subsume several known classes of valid inequalities for several special cases; on the other hand, they yield many new inequalities for several other special cases. For some of the classes we also prove a lower bound on their Gomory–Chvátal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering, as well as to balanced and ideal matrices.

### Citations

1459 | Theory of linear and integer programming - Schrijver - 1986 |

1038 | Integer and Combinatorial Optimization - Nemhauser, Wolsey - 1988 |

500 |
Graphs and Hypergraphs
- Berge
- 1973
(Show Context)
Citation Context ...the set of nodes, and 2C is a collection of distinct subsets of N, the set of (hyper)edges. We only deal with hypergraphs without loops, i. e., we always assume that HI > 2 for all H E -. We refer to =-=[Ber73]-=- for a thorough introduction to hypergraphs. Here, we are interested in hypergraphs with additional node subsets associated with each edge. Definition 2.1. Let (N, -C) be a hypergraph, and let tr: C -... |

223 | Combinatorial Optimization - Cook, Cunningham, et al. - 1998 |

143 | Edmonds polytopes and a hierarchy of combinatorial problems - Chvátal - 1973 |

108 | On the cut polytope - Barahona, Mahjoub - 1986 |

100 | Directed hypergraphs and applications
- Gallo, Longo, et al.
- 1993
(Show Context)
Citation Context ...odes and of a set of directed hyperedges (hyperarcs). A hyperarc (H + , H-) C SC consists of two (possibly empty) disjoint subsets of N. For a survey of directed hypergraphs the reader is referred to =-=[GLPN93]-=-. Now, consider for x {0, 1 }N the following "directed hypergraph covering" constraints x(H + ) + x(H-) > 1 for all hyperarcs (H+,H - ) E C, where x = - x is the complement of the 0/1 vector x. Observ... |

100 |
On the facial structure of set packing polyhedra
- Padberg
- 1973
(Show Context)
Citation Context ... only if u is isolated, i. e., if G does not contain an edge incident to u. This is a special case of the well-known fact that a clique inequality defines a facet if and only if the clique is maximal =-=[Pad73]-=-. Given a digraph D = (V,A) and an arc (u, v) E A, part (a) of Lemma 2.4 implies that xuv < 1 defines a facet of the acyclic subdigraph polytope of D if and only if (v, u) ' A. This was shown before b... |

79 | Set partitioning: A survey - Balas, Padberg - 1976 |

45 | The Boolean quadric polytope: some characteristics, facets and relatives - Padberg - 1989 |

44 | Geometric Algorithms and - Grötschel, Lovász, et al. - 1988 |

39 | Properties of vertex packing and independence system polyhedra - Nemhauser, Trotter - 1974 |

38 | A cutting plane algorithm for a clustering problem - Grötschel, Wakabayashi |

34 |
The partition problem
- Chopra, Rao
- 1993
(Show Context)
Citation Context ...revisit later. Indeed, besides those that can be interpreted as finding an independent set of maximum weight there are * the clique partitioning problem [GW89, GW90, ORS95], 2s* the partition problem =-=[CR93]-=-, * the transitive acyclic subdigraph problem [Mii196], * the interval graph completion problem [MS95, Sch96, MS99], * the relatively transitive subgraph problem [KL89, Sha89, SB]. One of our main pur... |

32 |
On cutting-plane proofs in combinatorial optimization. Linear Algebra and its Applications
- Chvátal, Cook, et al.
- 1989
(Show Context)
Citation Context ...r P has a cutting plane proof from Ax < b. Clearly, the length of a cutting plane proof of a valid inequality for P is at least its depth; however, the length can be significantly bigger (see, e. g., =-=[CCH89]-=-). The idea of deriving cutting planes by rounding based on exploitation of problem structure can in particular be used to obtain valid inequalities for the transitive packing polytope. Thereby, we al... |

30 |
Alpha-balanced graphs and matrices and GF(3)-representability of matroids
- Truemper
- 1982
(Show Context)
Citation Context ...in the corresponding fractional one. Recall that a 0/ 1 matrix is balanced if, in every submatrix with exactly two nonzero entries per row and per column, the sum of the entries is a multiple of four =-=[Tru82]-=-. We refer to Conforti, Cornujols, Kapoor, Vusk6vi6, and Rao [CCK+94] for a survey of balanced matrices and related concepts. Conforti and Comujols showed that a 0/ + 1 matrix A is balanced if and onl... |

29 | Decomposition of Balanced Matrices - Conforti, Cornuejols, et al. - 1999 |

28 | On cutting planes - Schrijver - 1980 |

25 | A class of facets producing graphs for vertex packing polyhedra - Trotter - 1975 |

24 |
Facets of the clique partitioning polytope
- Grötschel, Wakabayashi
(Show Context)
Citation Context ...d Reinelt [GJR85b]. If G is a graph without isolated edges, the assumption of part (b) of Lemma 2.4 is never met by an edge of the clique partitioning polytope of G. Indeed, Gr6tschel and Wakabayashi =-=[GW90]-=- proved that no upper bound constraint defines a facet of this polytope. Finally, part (b) of Lemma 2.4 tells us also that xe < 1 defines a facet of the partition polytope if the edge e does not belon... |

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

22 | On the acyclic subgraph polytope - Grötschel, Jünger, et al. - 1985 |

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

16 | The strongest facets of the acyclic subgraph polytope are unknown - Goemans, Hall - 1996 |

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

13 | PoIytopes and Scheduling - Schulz - 1995 |

12 |
Facets of the bipartite subgraph polytope
- Barahona, Grötschel, et al.
- 1985
(Show Context)
Citation Context ...ope, mainly based on the study of special configurations of the family of circuits. Among them are, to name a few, * the acyclic subdigraph polytope [GJR85b, Jin85], * the bipartite subgraph polytope =-=[BGM85]-=-, * the planar subgraph polytope [JM93]. We refer the reader to [EJR87, Lau89] and [BN89a, BN89b, CS89, NS89, Sas89] for the study of the facial structure of the independence system polytope in genera... |

11 | Set packing relaxations of some integer programs - Borndorfer, Weismantel - 2000 |

11 | On the 0–1 facets of the set covering polytope - Cornuéjols, Sassano - 1989 |

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

11 | On the partial order polytope of a digraph - Müller |

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 Cornu6jols 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. For instance, the inequalities for the set covering polytope associated with complete ... |

9 |
Covers and packings in a family of sets
- Edmonds
- 1962
(Show Context)
Citation Context ...alid when we allow for hypergraphs with loops. Then, we cover, for instance, the cut polytope (see, e.g., [BM86, DL97]) and the Boolean quadric polytope (e.g., [Pad89]) as well. It is well-known (see =-=[Edm62]-=-) that every set packing problem maximize cx subject to Ax < 1 (8.29) Xu E {0, 1} where A is a matrix of zeros and ones, can be transformed into an equivalent node packing problem on the intersection ... |

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

8 |
Clique-web facets for multicut polytopes
- Deza, Griitschel, et al.
- 1992
(Show Context)
Citation Context ...ties and the odd wheel inequalities for the clique partitioning and the partition polytope. The separation problem for the former class has previously been solved in [CF96, Mii193], the latter one in =-=[DGL92]-=-. For the odd partition inequalities we make use of both lower and upper bound constraints. Let us assume that H 1,H 2,... ,Hk is the underlying collection of edges and that d(u) odd implies that eith... |

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

8 | Solving the maximum weight planar subgraph problem by branch and cut
- Jünger, Mutzel
- 1993
(Show Context)
Citation Context ...l configurations of the family of circuits. Among them are, to name a few, * the acyclic subdigraph polytope [GJR85b, Jin85], * the bipartite subgraph polytope [BGM85], * the planar subgraph polytope =-=[JM93]-=-. We refer the reader to [EJR87, Lau89] and [BN89a, BN89b, CS89, NS89, Sas89] for the study of the facial structure of the independence system polytope in general. In Section 2, we introduce an extens... |

8 | On the Transitive Acyclic Subdigraph Polytope - Muller - 1993 |

7 | Geometric Algorithms and Combinatorial Optimization - Gr6tschel, Lovisz, et al. - 1988 |

7 | A cutting plane algorithm for a clustering problem - Gr6tschel, Wakabayashi - 1989 |

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 ...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 Cornu6jols 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. For instance, the inequalities for the set covering polyto... |

6 | Balanced 0, ±1-matrices, bicoloring and total dual integrality - Conforti, Cornuéjols - 1995 |

4 | Transitive packing
- Müller, Schulz
- 1996
(Show Context)
Citation Context ...spectively.) The node packing problem on a finite, undirected, loopless graph G with node weights is the problem of finding *This work is based on [Sch96, Chapter 4]; an extended abstract appeared in =-=[MS96]-=-. Most parts of this work were done while the second author was supported by the Deutsche Forschungsgemeinschaft via the graduate school "Computational Discrete Mathematics", grant We 1265/2-1. The fi... |

2 |
Edmonds polytopes and a hierarchy of combinatorial problems
- Chvital
- 1973
(Show Context)
Citation Context ...ch that aui > 1. The last inequality follows from XC2 + X{ui} E P, (N, , tr). O Theorem 4.12 was proven 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 the assumption of Theorem 4.12 is also satisfied by the k-fence inequalities since each dicycle containe... |

2 |
On the 0,1 facets of the set covering polytope
- ejols, Sassano
- 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 Cornu6jols 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. For instance, the inequalities ... |

2 |
Polyhedral results for antimatroids
- Korte, Lov'asz
- 1989
(Show Context)
Citation Context ...) - (7.28). If D is transitively closed, these points represent the independent sets of the transitivity antimatroid of D. Shallcross and Bland were motivated by a question raised by Korte and Lovisz =-=[KL89]-=- whether the convex hull of these incidence vectors has a (computationally) nice description. Shallcross and Bland present some conditions on D such that their polytope and therefore the relatively tr... |

2 | The interval order polytope of a digraph - Mueller - 1995 |

2 |
0� 1) ideal matrices
- Nobili, Sassano
- 1998
(Show Context)
Citation Context ...troduced by Conforti and Cornujols [CC95] in the context of balanced 0/ ± 1 matrices as the (fractional) generalized set covering polytope. Conforti and Cornujols [CC95] as well as Nobili and Sassano =-=[NS95]-=- 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 contribution to the stud... |

2 |
The clique partitioning polytope: Facets
- Oosten, Rutten, et al.
- 1995
(Show Context)
Citation Context ... incidence vectors of all clique partitionings in G. It was introduced and studied by Gr6tschel and Wakabayashi [GW89, GW90] and has recently been further investigated by Oosten, Rutten, and Spieksma =-=[ORS95]-=-. To show that it is an instance of a transitive packing polytope it is sufficient to deal with a graph instead of a hypergraph. Indeed, we take as the set N of nodes the edges of G, and two nodes are... |