## Local tree-width, excluded minors, and approximation algorithms

### Cached

### Download Links

- [www2.informatik.hu-berlin.de]
- [www.informatik.hu-berlin.de]
- DBLP

### Other Repositories/Bibliography

Venue: | Combinatorica |

Citations: | 55 - 6 self |

### BibTeX

@ARTICLE{Grohe_localtree-width,,

author = {Martin Grohe},

title = {Local tree-width, excluded minors, and approximation algorithms},

journal = {Combinatorica},

year = {},

volume = {23},

pages = {613--632}

}

### Years of Citing Articles

### OpenURL

### Abstract

ltw��Æ�Æ

### Citations

710 |
Graph Theory
- Diestel
- 2005
(Show Context)
Citation Context ...or every minor closed class C of graphs there is a finite set F of graphs such that C = {G | ∀H ∈ F : H �� G}. For a nice introduction to graph minor theory we refer the reader to the last chapter of =-=[7]-=-, a recent survey is [20]. Tree-decompositions. In this paper, we assume trees to be directed from the root to the leaves. If tu ∈ E T we call u a child of t and t the parent of u. The root of a tree ... |

388 | A separator theorem for planar graphs
- Lipton, Tarjan
- 1979
(Show Context)
Citation Context ...n 6). 1 Actually, the main result of Alon, Seymour, and Thomas’s article is a separator theorem for graphs with an excluded minor, generalizing a well-known separator theorem due to Lipton and Tarjan =-=[15]-=- for planar graphs. These separator theorems have numerous algorithmic applications, among them a polynomial time approximation scheme (PTAS) for the MAXIMUM INDEPENDENT SET problem on planar graphs [... |

307 |
Approximation algorithms for NP-complete problems on planar graphs
- Baker
- 1994
(Show Context)
Citation Context ...ilies of graphs of small tree-width in an orderly way. Such decompositions of planar graphs, better known under the name outerplanar decompositions, have been explored in various algorithmic settings =-=[5, 10, 14, 12]-=-. The main ideas go back to a fundamental article of Baker [5] on approximation algorithms on planar graphs. The local tree-width of a graph G = (V, E) is the function ltwG : N → N that associates wit... |

251 |
Graph Minors XIII. The Disjoint Paths Problem
- Robertson, Seymour
- 1995
(Show Context)
Citation Context ...also have to compute a tree-decomposition of a given graph overÄ���. Fortunately, Robertson and Seymour have proved another deep result that helps us with this task: Theorem 14 (Robertson and Seymour =-=[18]-=-). Every minor closed class of graphs has a polynomial time membership test. Lemma 15. Let�be a minor closed class of graphs. Then there is a polynomial time algorithm that computes, given a graph�, a... |

228 | Easy problems for tree-decomposable graphs
- Arnborg, Lagergren, et al.
- 1991
(Show Context)
Citation Context ...utions are sets X ⊆ V G such that for every edge vw ∈ E G either v ∈ X or w ∈ X (such sets X are called vertex covers), the cost function is defined by C(G, X) := |X|, and the goal is min. Lemma 5.2 (=-=[3]-=-). For every k ≥ 1, the restriction of MINIMUM VERTEX COVER to instances of tree-width at most k is solvable in linear time. Theorem 5.3. Let C be a class of graphs with an excluded minor. Then the re... |

161 |
Applications of a planar separator theorem
- Lipton, Tarjan
- 1980
(Show Context)
Citation Context ...] for planar graphs. These separator theorems have numerous algorithmic applications, among them a polynomial time approximation scheme (PTAS) for the MAXIMUM INDEPENDENT SET problem on planar graphs =-=[16]-=- and, more generally, classes of graphs with an excluded minor [1]. A different approach to approximation algorithms on planar graphs is Baker’s [5] technique based on the outerplanar decomposition. I... |

109 | Subgraph isomorphism in planar graphs and related problems, Information and
- Eppstein
- 1994
(Show Context)
Citation Context ...ilies of graphs of small tree-width in an orderly way. Such decompositions of planar graphs, better known under the name outerplanar decompositions, have been explored in various algorithmic settings =-=[5, 10, 14, 12]-=-. The main ideas go back to a fundamental article of Baker [5] on approximation algorithms on planar graphs. The local tree-width of a graph G = (V, E) is the function ltwG : N → N that associates wit... |

103 | Graph Minors XX. Wagner’s conjecture
- Robertson, Seymour
(Show Context)
Citation Context ...inor if, and only if, there is an n ≥ 1 such that C is Kn-free. Furthermore, this is equivalent to saying that C is contained in some non-trivial minor closed class of graphs. Robertson and Seymour’s =-=[18]-=- Graph Minor Theorem states that for every minor closed class C of graphs there is a finite set F of graphs such that C = {G | ∀H ∈ F : H �� G}. For a nice introduction to graph minor theory we refer ... |

93 | A separator theorem for graphs with an excluded minor and its applications
- Alon, Seymour, et al.
- 1990
(Show Context)
Citation Context ...hmic applications, among them a polynomial time approximation scheme (PTAS) for the MAXIMUM INDEPENDENT SET problem on planar graphs [16] and, more generally, classes of graphs with an excluded minor =-=[1]-=-. A different approach to approximation algorithms on planar graphs is Baker’s [5] technique based on the outerplanar decomposition. It does not only give another PTAS for MAXIMUM INDEPENDENT SET, but... |

85 |
Graph minors. III. Planar tree-width
- Robertson, Seymour
- 1984
(Show Context)
Citation Context ...mostÐ, for anÐ�. Then ÐÐ Ö for allÖÆ. The planar graph algorithms due to Baker and others that we mentioned in the introduction are based on the following result: Proposition 6 (Robertson and Seymour =-=[17]-=-). The class of planar graphs has linear local tree-width. More precisely, for every planar graph�andÖ�we have ltw�Ö�Ö. In this paper, a surface is a compact connected 2-manifold with (possibly empty)... |

84 | Diameter and treewidth in minor-closed graph families, Algorithmica 27
- Eppstein
- 2000
(Show Context)
Citation Context ...es of G and internally disjoint simple curves in S with the edges of G in such a way that a vertex v is incident with an edge e if, and only if, Π(v) is an endpoint of Π(e). Proposition 3.5 (Eppstein =-=[9]-=-). Let S be a surface. Then the class of all graphs embeddable in S has linear local tree-width. More precisely, there is a constant c such that for all graphs G embeddable in S and for all r ≥ 0 we h... |

74 | Deciding First-Order Properties of Locally TreeDecomposable Structures
- Frick, Grohe
(Show Context)
Citation Context ...ilies of graphs of small tree-width in an orderly way. Such decompositions of planar graphs, better known under the name outerplanar decompositions, have been explored in various algorithmic settings =-=[5, 10, 14, 12]-=-. The main ideas go back to a fundamental article of Baker [5] on approximation algorithms on planar graphs. The local tree-width of a graph G = (V, E) is the function ltwG : N → N that associates wit... |

61 |
A separator theorem for nonplanar graphs
- ALON, SEYMOUR, et al.
- 1990
(Show Context)
Citation Context ...Section 4 and turn to its applications now. In this paper, we focus on approximation algorithms. But let me mention that the theorem can also be used to re-prove a result of Alon, Seymour, and Thomas =-=[2]-=- that graphs G with an excluded minor have tree-width O( � |G|) (see Section 6). 1 Actually, the main result of Alon, Seymour, and Thomas’s article is a separator theorem for graphs with an excluded m... |

52 |
Graph Minors XVI. Excluding a Non-Planar Graph
- Robertson, Seymour
(Show Context)
Citation Context ...dmits such a decomposition, then C is not the class of all graphs.) The proof of this result is based on a deep structural characterization of graphs with excluded minors due to Robertson and Seymour =-=[17]-=-. We defer the precise technical statement of our decomposition theorem to Section 4 and turn to its applications now. In this paper, we focus on approximation algorithms. But let me mention that the ... |

48 | An approximation scheme for planar graph TSP
- Grigni, Koutsoupias, et al.
- 1995
(Show Context)
Citation Context ... would probably help to prove Theorem 4.2 directly without using Robertson’s and Seymour’s Theorem 4.1. The traveling salesman problem is another optimization problem that has a PTAS on planar graphs =-=[13, 4]-=-. It would be interesting to see if this problem has a PTAS on class of graphs with an excluded minor. Acknowledgements I thank Reinhard Diestel and Jörg Flum for helful comments on earlier versions o... |

35 | Towards a syntactic characterization of PTAS
- Khanna, Motwani
- 1996
(Show Context)
Citation Context |

29 | Fixed-parameter tractability, definability, and model checking
- Flum, Grohe
(Show Context)
Citation Context ...Seymour, Thomas [2]). Let�beÃÒ-free. Then tw��ÇÔ���. Ô����. Corollary 23. Let���Æand�Ä���. �Ø���Ø���Ý����Ü�� Deciding first-order properties. Another algorithmic application of Theorem 13 is given in =-=[10]-=-: It is proved that for every class�of graphs with an excluded minor there is a constant�such that for every property of graphs that is definable in first order logic there is anÇ���-algorithm decidin... |

27 | NC-algorithms for graphs with small treewidth
- Bodlaender
- 1989
(Show Context)
Citation Context .... Then ltw G (r) ≤ l(l − 1) r−1 for all r ∈ N. The planar graph algorithms due to Baker and others that we mentioned in the introduction are based on the following result: Proposition 3.4 (Bodlaender =-=[6]-=-). The class of planar graphs has linear local treewidth. More precisely, for every planar graph G and r ≥ 1 we have ltw G (r) ≤ 3r. In this paper, a surface is a compact connected 2-manifold with (po... |

10 |
Graph minors. XIII: the disjoint paths problem
- Roberson, Seymour
- 1995
(Show Context)
Citation Context ...have to compute a tree-decomposition of a given graph over L(λ, µ). Fortunately, Robertson and Seymour have proved another deep result that helps us with this task: Theorem 4.3 (Robertson and Seymour =-=[19]-=-). Every minor closed class of graphs has a polynomial time membership test. Lemma 4.4. Let C be a minor closed class of graphs. Then there is a polynomial time algorithm that computes, given a graph ... |

4 | Excluding a countable clique
- Diestel, Thomas
- 1999
(Show Context)
Citation Context ... [17]). For every n ∈ N there exist µ ∈ N and surfaces S, S ′ such that all Kn-free graphs have a tree-decomposition over A(S, µ) ∪ A(S ′ , µ). Further details concerning this theorem can be found in =-=[8, 20, 17]-=-. For λ, µ ≥ 0 we let L(λ) := � G � � � H ∀H � G ∀r ≥ 0 : ltw (r) ≤ λ · r , � � � L(λ, µ) := G � ∃X ⊆ V G : � �X� ≤ µ ∧ G \ X ∈ L(λ) �� . Note that L(λ, µ) is minor closed and that ω(L(λ, µ)) = λ + µ ... |

3 | Recent excluded minor theorems
- Thomas
- 1999
(Show Context)
Citation Context ...ass C of graphs there is a finite set F of graphs such that C = {G | ∀H ∈ F : H �� G}. For a nice introduction to graph minor theory we refer the reader to the last chapter of [7], a recent survey is =-=[20]-=-. Tree-decompositions. In this paper, we assume trees to be directed from the root to the leaves. If tu ∈ E T we call u a child of t and t the parent of u. The root of a tree T is always denoted by r ... |

2 |
A polynomialtime approximation scheme for weighted planar graph TSP
- Klein, Woloszyn
- 1998
(Show Context)
Citation Context ... would probably help to prove Theorem 4.2 directly without using Robertson’s and Seymour’s Theorem 4.1. The traveling salesman problem is another optimization problem that has a PTAS on planar graphs =-=[13, 4]-=-. It would be interesting to see if this problem has a PTAS on class of graphs with an excluded minor. Acknowledgements I thank Reinhard Diestel and Jörg Flum for helful comments on earlier versions o... |

2 | Fixed-parameter tractability and logic
- Flum, Grohe
- 1999
(Show Context)
Citation Context ... and obtain tw(G) ≤ 3 � λ · |G|. ✷ Corollary 6.2. Let λ, µ ∈ N and G ∈ L(λ, µ). Then tw(G) ≤ 3 � λ|G| + µ. Corollary 6.3. Let G be Kn-free. Then tw(G) ≤ O( � |G|). Deciding first-order properties. In =-=[11]-=- we give another algorithmic application of Theorem 4.2. We show that for every class C of graphs with an excluded minor there is a constant c > 0 such that for every property of graphs that is defina... |