## The symmetric traveling salesman polytope: New facets from the graphical relaxation (2007)

Venue: | MATHEMATICS OF OPERATIONS RESEARCH |

Citations: | 10 - 0 self |

### BibTeX

@TECHREPORT{Naddef07thesymmetric,

author = {Denis Naddef and Giovanni Rinaldi},

title = {The symmetric traveling salesman polytope: New facets from the graphical relaxation},

institution = {MATHEMATICS OF OPERATIONS RESEARCH},

year = {2007}

}

### OpenURL

### Abstract

### Citations

127 |
The Traveling Salesman Problem
- Jünger, Rinaldi
- 1997
(Show Context)
Citation Context ...mplete, and it is very unlikely that it will ever be. In the last 40 years many papers appeared in which new valid or facetdefining inequalities for STSP�n� were introduced. We refer to Jünger et al. =-=[12]-=-, Lawler et al. [13], Naddef [14], and Naddef and Pochet [15] for a list of them and for further details on the traveling salesman polytope. As new inequalities are discovered, it becomes more and mor... |

56 |
The traveling salesman problem on a graph and some related integer polyhedra
- Cornuéjols, Fonlupt, et al.
- 1985
(Show Context)
Citation Context ...ostly to the exploitation of the current (partial) knowledge of the structure of the traveling salesman polytope and is the main motivation for pushing this knowledge a bit further. Cornuéjols et al. =-=[5]-=- define a class of valid inequalities for the symmetric traveling salesman polytope on a graph with n nodes �STSP�n�� known as the path, the wheelbarrow, and the bicycle inequalities (the PWB inequali... |

36 |
Edmonds polytopes and weakly Hamiltonian graphs
- Chvátal
- 1973
(Show Context)
Citation Context ...rations Research 32(1), pp. 233–256, © 2007 INFORMS H T 1 T 2 T 3 2 1 1 3 1 1 2 1 2 1 1 2 1 1 Figure 2. A 3-tooth comb and the coefficients of the inequality in TT form. When first defined by Chvátal =-=[3]-=- and then by Grötschel and Padberg [9], the inequality, where k is odd and stands for the number of teeth (in our example k = 3) and where ��U� denotes the edge set ��u� w� � u� w ∈ U�, was given as k... |

34 |
On the symmetric travelling salesman problem I: Inequalities
- Grötschel, Padberg
- 1979
(Show Context)
Citation Context ...that the comb inequalities define facets of STSP�n�. Another such proof for comb inequalities can be found in Naddef and Wild [21], and also, of course, in the original proof of Grötschel and Padberg =-=[9, 10]-=-. 233sNaddef and Rinaldi: The Symmetric Traveling Salesman Polytope: New Facets from the Graphical Relaxation 234 Mathematics of Operations Research 32(1), pp. 233–256, © 2007 INFORMS In Naddef and Ri... |

33 |
On the symmetric travelling salesman problem II: Lifting theorems and facets.Math
- Grötschel, Padberg
- 1979
(Show Context)
Citation Context ...that the comb inequalities define facets of STSP�n�. Another such proof for comb inequalities can be found in Naddef and Wild [21], and also, of course, in the original proof of Grötschel and Padberg =-=[9, 10]-=-. 233sNaddef and Rinaldi: The Symmetric Traveling Salesman Polytope: New Facets from the Graphical Relaxation 234 Mathematics of Operations Research 32(1), pp. 233–256, © 2007 INFORMS In Naddef and Ri... |

30 |
On the symmetric travelling salesman problem: a computational study
- Padberg, Hong
- 1980
(Show Context)
Citation Context ...ch the application of such operation preserves the facet-defining property of an inequality. Here we apply this operation to the PWB inequalities. The so-called chain inequalities of Padberg and Hong =-=[24]-=- are a particular case of the inequalities obtained in this way. We finally prove that the repeated 2-sum composition of PWB inequalities yields facet-defining inequalities. Clique trees, with at leas... |

29 |
Clique tree inequalities and the symmetric travelling salesman problem
- GROTSCHEL, PULLEYBLANK
- 1986
(Show Context)
Citation Context ...e outside all handles and teeth, are a special case of these inequalities. Therefore, we get an alternative proof that they are facet-defining; the original one was given by Grötschel and Pulleyblank =-=[11]-=-. Let G = �V � E� be a graph on n nodes. By e = �u� v� we denote the edge of G having u and v as end nodes, and we let � E be the set of all real vectors whose components are indexed by the edge set E... |

28 | Worst-case Comparison of Valid Inequalities for the TSP - Goemans - 1995 |

27 |
Combinatorial optimization and small polytopes
- Christof, Reinelt
- 1996
(Show Context)
Citation Context ... While for small values of n (n ≤ 9) a complete description of the system of inequalities defining facets of STSP�n� has been generated by means of a computer program (see, e.g., Christof and Reinelt =-=[2]-=-), for arbitrary values of n the knowledge of such a system is far from being complete, and it is very unlikely that it will ever be. In the last 40 years many papers appeared in which new valid or fa... |

27 |
The graphical relaxation: A new framework for the symmetric travelling salesman polytope
- Naddef, Rinaldi
- 1993
(Show Context)
Citation Context ...2 ∈ �c2�v2�. Before stating the main theorem, we prove a lemma concerning the h-liftability of simple regular parity PWB-tree inequalities and recall three lemmata that we state in Naddef and Rinaldi =-=[18]-=- and that will be used in the main proof as well. Lemma 5.1. The 2-sum of a simple regular parity PWB-tree and of a simple PWB inequality is h-liftable. Proof. We consider the 2-sum of a simple regula... |

26 |
The symmetric travelling salesman polytope and its graphical relaxation: Composition of valid inequalities", Mathematical Programming 51,359400
- Naddef, Rinaldi
- 1991
(Show Context)
Citation Context ...equalities for GTSP�n� are also facet-defining for STSP�n�. Moreover, we state sufficient conditions under which the composition of facetdefining inequalities that we introduced in Naddef and Rinaldi =-=[17]-=- for GTSP�n� also applies to the case of STSP�n�. Finally, we show how to apply several kinds of liftings to the inequalities that define facets of STSP�n�. In this paper we first use the technique of... |

21 | Combinatorial optimization
- Grotschel, Lovasz
- 1995
(Show Context)
Citation Context ...success of the cutting plane technique has certainly been achieved for the traveling salesman problem” Grötschel and Lovász observe in their survey on combinatorial optimization (Grötschel and Lovász =-=[8]-=-). This success is due mostly to the exploitation of the current (partial) knowledge of the structure of the traveling salesman polytope and is the main motivation for pushing this knowledge a bit fur... |

9 |
Using path inequalities in a branch and cut code for the symmetric traveling salesman problem
- Clochard, Naddef
- 1993
(Show Context)
Citation Context ...es from the fact that they are useful in a polyhedral cutting plane algorithm for the solution of the traveling salesman problem, as shown by Naddef and Thienel [19, 20] (see also Clochard and Naddef =-=[4]-=-). This experimental evidence confirms the theoretical expectation expressed by Goemans [6], where he addresses the problem of measuring the quality of a class of inequalities. As a measure, he propos... |

5 | The symmetric traveling salesman polytope revisited
- Naddef, Pochet
- 2001
(Show Context)
Citation Context ...last 40 years many papers appeared in which new valid or facetdefining inequalities for STSP�n� were introduced. We refer to Jünger et al. [12], Lawler et al. [13], Naddef [14], and Naddef and Pochet =-=[15]-=- for a list of them and for further details on the traveling salesman polytope. As new inequalities are discovered, it becomes more and more difficult to keep track of all of them in a unifying framew... |

3 |
On the monotonization of polyhedra
- Balas, Fischetti
- 1997
(Show Context)
Citation Context ...], and the graphical traveling salesman polyhedron. Sufficient conditions for a facet-defining inequality for the monotone relaxation to be facet-defining for STSP�n� are given by Balas and Fischetti =-=[1]-=-. A desirable property of a relaxation R is that every facet of STSP�n� be contained in exactly one of the facets of R that do not contain the entire polytope STSP�n�. If this property holds, then the... |

3 |
On the monotone symmetric travelling salesman problem: hypohamiltonian/hypotraceable graphs and facets
- Grotschel
- 1980
(Show Context)
Citation Context ...h a property for STSP�n�. The two major relaxations that have been considered in the study of the polyhedral structure of STSP�n� are the monotone traveling salesman polytope, introduced by Grötschel =-=[7]-=-, and the graphical traveling salesman polyhedron. Sufficient conditions for a facet-defining inequality for the monotone relaxation to be facet-defining for STSP�n� are given by Balas and Fischetti [... |

3 | Not every GTSP facet induces an STSP facet
- Oswald, Reinelt, et al.
- 2005
(Show Context)
Citation Context ...at: 1 Until very recently, no examples were known of facet-defining inequalities for GTSP�n� that are provably not facet-defining for STSP�n�. Some of such inequalities are exhibited by Oswald et al. =-=[23]-=-.sNaddef and Rinaldi: The Symmetric Traveling Salesman Polytope: New Facets from the Graphical Relaxation Mathematics of Operations Research 32(1), pp. 233–256, © 2007 INFORMS 237 (a) c e = 0 for all ... |

1 |
Polyhedral theory, branch and cut algorithms for the symmetric traveling salesman problem
- Naddef
(Show Context)
Citation Context ...hat it will ever be. In the last 40 years many papers appeared in which new valid or facetdefining inequalities for STSP�n� were introduced. We refer to Jünger et al. [12], Lawler et al. [13], Naddef =-=[14]-=-, and Naddef and Pochet [15] for a list of them and for further details on the traveling salesman polytope. As new inequalities are discovered, it becomes more and more difficult to keep track of all ... |