## On the Graphical Relaxation of the Symmetric Traveling Salesman (2007)

Citations: | 3 - 3 self |

### BibTeX

@MISC{Oswald07onthe,

author = {Marcus Oswald and Gerhard Reinelt and Dirk Oliver Theis},

title = {On the Graphical Relaxation of the Symmetric Traveling Salesman},

year = {2007}

}

### OpenURL

### Abstract

The Graphical Traveling Salesman Polyhedron (GTSP) has been proposed by Naddef and Rinaldi to be viewed as a relaxation of the Symmetric Traveling Salesman Polytope (STSP). It has also been employed by Applegate, Bixby, Chvátal, and Cook for solving the latter to optimality by the Branch-and-Cut method. There is a close natural connection between the two polyhedra. Until now, it was not known whether there are facets in TT-form of the GTSP polyhedron which are not facets of the STSP polytope as well. In this paper we give an affirmative answer to this question for n ≥ 9. We provide a general method for proving the existence of such facets, at the core of which lies the construction of a continuous curve on a polyhedron. This curve starts in a vertex, walks along edges, and ends in a vertex not adjacent to the starting vertex. Thus there must have been a third vertex on the way.

### Citations

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(Show Context)
Citation Context ...of the graphical relaxation increases with our findings. The paper concludes with an outlook in Section 6. 2. Preliminaries We assume knowledge of the basics of polyhedral theory (as outlined e.g. in =-=[24]-=-) and elementary facts for blocking type polyhedra. Recall that for a polyhedron P ⊆sm of blocking type, the blocking polyhedron B(P ) is defined as B(P ) := {a ∈sm + | ∀x ∈ P : ax ≥ 1}. For a ∈sm and... |

176 | On the solution of traveling salesman problems
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Citation Context ...t they also define facets of STSP(n). More relevant to the practical solution of the STSP is the contribution of Applegate, Bixby, Chvátal and Cook: their “local cuts” separation method [3] (see also =-=[2,4]-=-) explicitly relies on the Graphical Relaxation and was a key component for solving STSP instances of more than 10000 cities to optimality. The method produces inequalities which are valid and facet-d... |

127 |
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(Show Context)
Citation Context ...his polytope is at the heart of the well-known branch-and-cut approach to solve the symmetric TSP [13] to optimality and knowledge about its facet structure allows for the solution of large instances =-=[4,19]-=-. A variant of the classical TSP is the Graphical Traveling Salesman Problem (GTSP). Here the underlying graph is not necessarily complete and the task is to find a closed walk of minimum length visit... |

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Citation Context ...ph Kn = (Vn, En), where Vn := {0, . . . , n − 1} and En is the set of all two-element subsets of Vn. This polytope is at the heart of the well-known branch-and-cut approach to solve the symmetric TSP =-=[13]-=- to optimality and knowledge about its facet structure allows for the solution of large instances [4,19]. A variant of the classical TSP is the Graphical Traveling Salesman Problem (GTSP). Here the un... |

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Citation Context ...between cities are allowed. As a consequence, some edges may have to be used more than once. The study of the Graphical Traveling Salesman Problem has been initiated by Cornuéjols, Fonlupt and Naddef =-=[12]-=- and Fleischmann [14]. An interesting current line of research is to try to exploit special properties (e.g., planarity) of the underlying sparse graphs. University of Heidelberg, Im Neuenheimer Feld ... |

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Citation Context ...he only non-NR facet. We only sketch a relationship between sets of facets which, when interpreted as an LP-relaxation of GTSP(n), satisfy the so-called parsimonious property of Goemans and Bertsimas =-=[17]-=- and the structure of the ridge graph of GTSP(n). Recall that the ridge graph of a polyhedron P has as its node set the set of all facets of P , and as its edge set the set of ridges of P . An LP-rela... |

37 | Implementing the dantzigfulkerson-johnson algorithm for large traveling salesman problems
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Citation Context ...his polytope is at the heart of the well-known branch-and-cut approach to solve the symmetric TSP [13] to optimality and knowledge about its facet structure allows for the solution of large instances =-=[4,19]-=-. A variant of the classical TSP is the Graphical Traveling Salesman Problem (GTSP). Here the underlying graph is not necessarily complete and the task is to find a closed walk of minimum length visit... |

34 |
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Citation Context ...r an inequality (a, α), we call (a, α) − � u ¯ λu(a) (du, 1) its TT-form representative. We repeat some facts from the literature. The first one probably has to be attributed to Grötschel and Padberg =-=[18]-=- or Maurras [20], the remaining two have been proven in [22]. a. The degree equations x(δ(u)) = 2, u ∈ Vn, form a complete system of equations for STSP(n). (Therefore, if two inequalities differ by a ... |

27 |
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Citation Context ...U ⊆ Vn, we denote by δ(U) the set of all edges of Kn with precisely one end node in U, and we define δ(u) := δ({u}). We also write x(F ) := � e∈F xe for a set of edges F . In their seminal 1993 paper =-=[22]-=-, starting with the observation that STSP(n) is a face of the Graphical Traveling Salesman Polyhedron GTSP(n) for the complete graph Kn, Naddef and Rinaldi develop a machinery to deal with the polyhed... |

26 | TSP Cuts Which Do Not Conform to the Template Paradigm
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Citation Context ...is implies that they also define facets of STSP(n). More relevant to the practical solution of the STSP is the contribution of Applegate, Bixby, Chvátal and Cook: their “local cuts” separation method =-=[3]-=- (see also [2,4]) explicitly relies on the Graphical Relaxation and was a key component for solving STSP instances of more than 10000 cities to optimality. The method produces inequalities which are v... |

25 |
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Citation Context ...d H must be TT-disjoint in at least one node, i.e., λ(µ) �= 0, for all µ. u=0 u=0sOn the Graphical Relaxation of the STSP 7 Lemma 2. γµ never vanishes, i.e., for all µ ≥ 0 we have γµ > 0. Proof. From =-=[21]-=- (or from the fact that GTSP(n) is of blocking type) it is known that the only facets of GTSP(n) which are defined by inequalities whose right hand side vanishes, are the non-negativity facets. Hence,... |

21 |
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Citation Context ...o Lemma 4). We do not know of any result on complete descriptions for GTSP(n) for n ≥ 6, either theoretically or computationally. The complete description of STSP(n) for n = 6 and n = 7 was proven in =-=[5]-=-, for n = 8 it was computed in [8], and for n = 9 it was computed in [11]. For n = 10 the same system of inequalities was produced in [1,11], and it is conjectured to be complete in [11]. We used the ... |

14 |
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Citation Context ...result on complete descriptions for GTSP(n) for n ≥ 6, either theoretically or computationally. The complete description of STSP(n) for n = 6 and n = 7 was proven in [5], for n = 8 it was computed in =-=[8]-=-, and for n = 9 it was computed in [11]. For n = 10 the same system of inequalities was produced in [1,11], and it is conjectured to be complete in [11]. We used the following method to verify computa... |

14 | Reinelt: Decomposition and Parallelization Techniques for Enumerating the Facets of 0/1-Polytopes
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(Show Context)
Citation Context ...orm inequalities: there are only n(n + 1)/2 facets which are not defined by TT-form inequalities, but it is known that the total number of facets of, for example, GTSP(10), is at least 51,043,900,866 =-=[1,11]-=-. While Naddef and Rinaldi did not conjecture an answer to (∗), others have conjectured a yes-answer [7]. Being only of theoretical interest in the beginning, this question recently has gained increas... |

11 |
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(Show Context)
Citation Context ...ps valid inequalities defining ridges of P which are not contained in non-negativity facets to points on bounded edges of B(P ). Let E(n) denote the set of vertices of GTSP(n). It is well-known (e.g. =-=[16]-=-) that GTSP(n) is of blocking type. We write E 0 (n) for the set of vertices of GTSP(n) which represent Hamiltonian cycles. These are precisely the vertices of STSP(n), and satisfy |x| 1 := � e xe = n... |

9 |
The traveling salesman problem in graphs with 3-edge cutsets
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(Show Context)
Citation Context ...ts G to H and walks along edges of the blocking polyhedron. The points represent vertices of B(GTSP(n)). In the hollow points, (φ(s), 1) defines a non-NR-facet of GTSP(n). 4. Examples It follows from =-=[15]-=- that the answer to (∗) is yes for n ≤ 5. In this section we show that this is also true for 6 ≤ n ≤ 8. After that we apply the results of the previous section to introduce a non-NR-facet for n = 9. E... |

7 |
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(Show Context)
Citation Context ...ng, this question recently has gained increasing practical importance. The Graphical Relaxation was not only used to obtain theoretical results for the STSP, but also for cutting plane generation. In =-=[6,7]-=-, a separation method for STSP is introduced which produces inequalities that define facets of GTSP(n), and in [7] it is conjectured that this implies that they also define facets of STSP(n). More rel... |

7 |
Separation algorithms for classes of STSP inequalities arising from a new STSP relaxation
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(Show Context)
Citation Context ... known that the total number of facets of, for example, GTSP(10), is at least 51,043,900,866 [1,11]. While Naddef and Rinaldi did not conjecture an answer to (∗), others have conjectured a yes-answer =-=[7]-=-. Being only of theoretical interest in the beginning, this question recently has gained increasing practical importance. The Graphical Relaxation was not only used to obtain theoretical results for t... |

5 |
A cutting plane procedure for travelling salesman problem on road networks
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- 1985
(Show Context)
Citation Context ...lowed. As a consequence, some edges may have to be used more than once. The study of the Graphical Traveling Salesman Problem has been initiated by Cornuéjols, Fonlupt and Naddef [12] and Fleischmann =-=[14]-=-. An interesting current line of research is to try to exploit special properties (e.g., planarity) of the underlying sparse graphs. University of Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg... |

5 |
Some results on the convex hull of Hamiltonian cycles of symmetric complete graphs
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(Show Context)
Citation Context ...(a, α), we call (a, α) − � u ¯ λu(a) (du, 1) its TT-form representative. We repeat some facts from the literature. The first one probably has to be attributed to Grötschel and Padberg [18] or Maurras =-=[20]-=-, the remaining two have been proven in [22]. a. The degree equations x(δ(u)) = 2, u ∈ Vn, form a complete system of equations for STSP(n). (Therefore, if two inequalities differ by a linear combinati... |

3 |
Hamiltonian path and symmetric travelling salesman polytopes
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(Show Context)
Citation Context ... does the 0-node lifted inequality (a ◦ , α) define a facet of STSP(n + 1)? In [22], sufficient conditions on NR-facets of GTSP(n) were given, which imply that the 0-node lifted facet is still NR. In =-=[23]-=-, it was shown that these conditions are also necessary. Further, [23] proves that if an NR-facet is 0-node lifted twice at the same node, then the result is again an NR-facet. It is possible to exten... |

2 |
Linear optimization and extensions: problems and solutions
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(Show Context)
Citation Context ...orm inequalities: there are only n(n + 1)/2 facets which are not defined by TT-form inequalities, but it is known that the total number of facets of, for example, GTSP(10), is at least 51,043,900,866 =-=[1,11]-=-. While Naddef and Rinaldi did not conjecture an answer to (∗), others have conjectured a yes-answer [7]. Being only of theoretical interest in the beginning, this question recently has gained increas... |

2 |
PORTA – a polyhedron representation algorithm
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(Show Context)
Citation Context ...5676 7 360 4770 111982 8 2520 42840 ≥2669752 9 20160 428400 ≥74896444 10 181440 4717440 ≥2388208240 Table 1. Number of vertices of STSP(n) and GTSP(n). We applied the facet enumeration software PORTA =-=[9]-=- in a standard way to check the correctness of the conjecture for n = 6 by computing all facets of GTSP(6). However, the computation did not terminate within 50 days and we had to give up. The numbers... |

2 |
SmaPo – library of Small Polytopes. http://www. informatik.uni-heidelberg.de/groups/comopt/software/SMAPO/tsp/tsp.html
- Christof, Reinelt
(Show Context)
Citation Context ...ontains two ridges: a facet. Let (˙a, 18) and ( ˙ b, 28) be the two inequalities whose coefficients are displayed in Table 2. They are taken from the list of facets of STSP(n), n = 6, . . . , 10 (see =-=[10]-=-). Since they are in TT-form, they define NR-facets by item (b) in Section 2. By enumerating all x ∈ E 0 (9) it is easy to see that they are adjacent on STSP(9). They are TT-disjoint in node 0 but now... |

2 |
Polyhedra and Algorithms for the General Routing Problem
- Theis
- 2005
(Show Context)
Citation Context ...aths between their extremities using the shown edges. The right hand side is 44. With computer aid, and relying on so-called “tilting complexes”, which we do not have the space to introduce here (see =-=[25]-=-), it is possible to prove that, for n = 9, this non-NR example is, except for permutation of the nodes, the only non-NR facet. We only sketch a relationship between sets of facets which, when interpr... |