## Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring (2005)

Venue: | In 46th Annual IEEE Symposium on Foundations of Computer Science |

Citations: | 43 - 12 self |

### BibTeX

@INPROCEEDINGS{Demaine05algorithmicgraph,

author = {Erik D. Demaine and Mohammadtaghi Hajiaghayi and Ken-ichi Kawarabayashi},

title = {Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring},

booktitle = {In 46th Annual IEEE Symposium on Foundations of Computer Science},

year = {2005},

pages = {637--646},

publisher = {Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.

### Citations

364 |
Graphs minors II. Algorithmic aspects of tree-width
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...enerally, we use the term “H-minor-free” to refer to any minor-closed graph class that excludes some fixed graph H. Second we define the basic notion of treewidth, introduced by Robertson and Seymour =-=[40]-=-. To define this notion, first we consider a representation of a graph as a tree, called a tree decomposition. Precisely, a tree decomposition of a graph G = (V, E) is a pair (T, χ) in which T = (I, F... |

307 |
Approximation algorithms for NP-complete problems on planar graphs
- Baker
- 1994
(Show Context)
Citation Context ...ing PTASs for a variety of minimum and maximization problems, essentially providing a generalized Baker’s approach that applies to all H-minor-free graphs, not just apex-minor-free (or planar) graphs =-=[3, 24]-=-. Our approximations to treewidth and grid minors exploit the minimax relation between these two quanties, leading to a combinatorial “primal-dual” type algorithm. This approach also leads us to effic... |

275 |
Every planar map is four-colorable
- Appel, Haken
- 1976
(Show Context)
Citation Context ... with |V (H)|−1 colors. Hadwiger [33] posed this problem in 1943, and proved the conjecture for |V (H)| ≤ 4. The case |V (H)| = 5 is equivalent to the four-color theorem [52], and therefore also true =-=[2, 1, 38]-=-. The case |V (H)| = 6 was proved by Robertson, Seymour, and Thomas [39], also using the four-color theorem. All cases |V (H)| ≥ 7 remain unsolved. The best general upper bound is that every H-minor-f... |

176 | Zero knowledge and the chromatic number
- Feige, Kilian
- 1996
(Show Context)
Citation Context ... as minimum chromatic number) in H-minor-free graphs. Graph coloring is one of the hardest problems to approximate: in general graphs, it is inapproximable within n 1−ε for any ε > 0, unless ZPP = NP =-=[26]-=-. Even for 3-colorable graphs, the best approximation algorithm achieves a factor of O(n 3/14 lg O(1) n) [5]. In planar graphs, the problem is 4/3-approximable, and that is the best possible unless P ... |

163 | Graph Minors X. Obstructions to Treedecomposition - Robertson, Seymour - 1991 |

160 | The dense k-subgraph problem
- Feige, Kortsarz, et al.
(Show Context)
Citation Context ...same technique as coloring can be applied to many other problems. An example of a maximization problem is the notorious dense k-subgraph problem, for which the best approximation known is O(n 1/3−ε ) =-=[27]-=-. We obtain a 2-approximation for H-minor-free graphs. 3.2. Approximation Algorithms for Minimization Problems We start with a simple but very general constant-factor approximation, which in some case... |

113 | The four-colour theorem
- Robertson, Sanders, et al.
- 1997
(Show Context)
Citation Context ... with |V (H)|−1 colors. Hadwiger [33] posed this problem in 1943, and proved the conjecture for |V (H)| ≤ 4. The case |V (H)| = 5 is equivalent to the four-color theorem [52], and therefore also true =-=[2, 1, 38]-=-. The case |V (H)| = 6 was proved by Robertson, Seymour, and Thomas [39], also using the four-color theorem. All cases |V (H)| ≥ 7 remain unsolved. The best general upper bound is that every H-minor-f... |

110 |
Quickly excluding a planar graph
- Robertson, Seymour, et al.
- 1994
(Show Context)
Citation Context ...]. We sketch how to obtain the structure in the first component. If the treewidth of the graph is small, the problem is relatively easy, so we focus on the case of large treewidth. The main result of =-=[48]-=- says that there exists a function f(k) such that, if the treewidth of a graph G is at least f(k), then G has a k × k grid minor. For our algorithmic purposes, we shall use a k-wall instead of a k × k... |

104 | Excluded minors, network decomposition, and multicommodity flow
- Klein, Plotkin, et al.
- 1993
(Show Context)
Citation Context ...lements of the set, the cardinality of the set, or even the order of the largest graph in the set” [28]. Algorithms for H-minor-free graphs for a fixed graph H have been studied extensively; see e.g. =-=[8, 31, 9, 35, 37]-=-. In particular, it is generally believed that several algorithms for planar graphs can be generalized to H-minor-free graphs for any fixed H [31, 35, 37]. The decomposition theorem provides the key i... |

88 |
Call routing and the ratcatcher
- Seymour, Thomas
- 1994
(Show Context)
Citation Context ...d graphs are planar. Because we only remove vertices, the treewidth of each reduced graph is at most tw(Bi). Now we can apply a 1.5-approximation for minimum-width tree decomposition on planar graphs =-=[49]-=- to obtain a tree decomposition of each reduced graph with width O(tw(Bi)). Now we reverse the noose-removal process, in each round adding the removed nooses from each of the reduced graphs, and recom... |

84 | Diameter and treewidth in minor-closed graph families, Algorithmica 27
- Eppstein
- 2000
(Show Context)
Citation Context ...ing PTASs for a variety of minimum and maximization problems, essentially providing a generalized Baker’s approach that applies to all H-minor-free graphs, not just apex-minor-free (or planar) graphs =-=[3, 24]-=-. Our approximations to treewidth and grid minors exploit the minimax relation between these two quanties, leading to a combinatorial “primal-dual” type algorithm. This approach also leads us to effic... |

72 |
Finding a maximum cut of a planar graph in polynomial time
- Hadlock
- 1975
(Show Context)
Citation Context ...s equivalent to EMISP(π) where π is the property of the graph being bipartite; our PTAS for this problem is an interesting complement to the polynomial-time algorithm for maximum cut in planar graphs =-=[32]-=-. Theorem 3.7 For any hereditary graph property π that can be solved in polynomial time on graphs of bounded treewidth, for any graph H, and for any ε > 0, there is a polynomial-time (1 + ε)-approxima... |

68 | edge-deletion NP-complete problems, in
- Yannakakis, Node-
- 1978
(Show Context)
Citation Context ...s include finding the maximum induced subgraph that is chordal, acyclic, without cycles of a specified length, without edges (independent set), of maximum degree r ≥ 1, bipartite, a clique, or planar =-=[53]-=-. Yannakakis [53] has shown that these examples of MISP are all NP-complete, and for all except the last example, NP-complete even when restricted to planar graphs [53]. Another interesting example is... |

65 |
Über eine Klassifikation der Streckenkomplexe
- Hadwiger
- 1943
(Show Context)
Citation Context ...onnections to Hadwiger’s conjecture, one of the major unsolved problems in graph theory, which can be stated as follows: every H-minor-free graph has a vertex coloring with |V (H)|−1 colors. Hadwiger =-=[33]-=- posed this problem in 1943, and proved the conjecture for |V (H)| ≤ 4. The case |V (H)| = 5 is equivalent to the four-color theorem [52], and therefore also true [2, 1, 38]. The case |V (H)| = 6 was ... |

64 | On chromatic sums and distributed resource allocation
- Bar-Noy, Bellare, et al.
- 1998
(Show Context)
Citation Context ...to a wide variety of problems to which Baker’s approach applies, such as vertex cover. One particularly interesting application is the wellstudied variation of graph coloring called minimum color sum =-=[4, 26, 34]-=-, where the goal is to find a (vertex or edge) coloring with positive integers with minimum total value. We can use the bounded-treewidth algorithm of Halldórsson and Kortsarz [34], and the merging st... |

64 | Improved approximation algorithms for minimum-weight vertex separators
- Feige, Hajiaghayi, et al.
- 2008
(Show Context)
Citation Context ...mposition of width O(maxi tw(Pi)) = O(tw(G)). ✷ Recently, a noncombinatorial approach based on linear programming has been developed to obtain an O(1)approximation to treewidth in H-minor-free graphs =-=[25]-=-. 3.6. Approximating Grid Minors The primal-dual nature of the algorithm from the previous section allows us to construct grid minors as follows. The following result improves the fixed-parameter algo... |

55 |
Nonconstructive tools for proving polynomial-time decidability
- Fellows, Langston
- 1988
(Show Context)
Citation Context ...any minor-closed graph property. This consequence has been used to show the existence of polynomial-time algorithms for several graph problems, some of which were not previously known to be decidable =-=[28]-=-. However, these algorithmic results (except the minor test) are nonconstructive: we know that efficient algorithms exist, but do not know what they are. The difficulty is in determining the finite se... |

55 | Local Tree-width, Excluded Minors and Approximation Algorithms
- Grohe
(Show Context)
Citation Context ...ial form, the graph-minor decomposition theorem has already been used to obtain many combinatorial results and the existence of many efficient algorithms, despite being published only recently. Grohe =-=[30]-=- proves the existence of PTASs for minimum vertex cover, minimum dominating set, and maximum independent set in H-minor-free graphs. This theorem is existential not because the algorithm requires a d... |

53 |
The extremal function for complete minors
- Thomason
(Show Context)
Citation Context ...s that every H-minor-free graph has a vertex coloring with O(|V (H)| � lg |V (H)|) colors, which follows immediately from bounds on the average degree of a vertex in an H-minor-free graph; see, e.g., =-=[36, 51]-=-. Thus, Hadwinger’s conjecture is not resolved even up to constant factors, and the conjecture itself is only a worst-case bound. In contrast, our 2-approximation algorithm gives the best coloring, u... |

46 | Lower bound of the Hadwiger number of graphs by their average degree
- Kostochka
- 1984
(Show Context)
Citation Context ...s that every H-minor-free graph has a vertex coloring with O(|V (H)| � lg |V (H)|) colors, which follows immediately from bounds on the average degree of a vertex in an H-minor-free graph; see, e.g., =-=[36, 51]-=-. Thus, Hadwinger’s conjecture is not resolved even up to constant factors, and the conjecture itself is only a worst-case bound. In contrast, our 2-approximation algorithm gives the best coloring, u... |

43 | An Õ(n3/14 )-coloring algorithm for 3-colorable graphs
- Blum, Karger
- 1997
(Show Context)
Citation Context ...imate: in general graphs, it is inapproximable within n 1−ε for any ε > 0, unless ZPP = NP [26]. Even for 3-colorable graphs, the best approximation algorithm achieves a factor of O(n 3/14 lg O(1) n) =-=[5]-=-. In planar graphs, the problem is 4/3-approximable, and that is the best possible unless P = NP, essentially because all planar graphs are 4-colorable. In contrast, H-minor-free graphs (or even bound... |

43 |
Graph minors. VII. Disjoint paths on a surface
- Robertson, Seymour
- 1988
(Show Context)
Citation Context ...he large face-width. Our initial pair is (H0, Σ0) where Σ0 is the sphere. Every graph with high face-width that can be drawn in a surface in which H can also be drawn has an H-minor, by the result of =-=[41]-=-. Hence the genus-increasing process stops in O(|H| 2 ) steps or so. How does the rest of the graph attach to Hi? It turns out that, except for a bounded number of disks which become vortices, the res... |

40 | Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs
- Demaine, Fomin, et al.
(Show Context)
Citation Context ...tive as mentioned above). We can modify the algorithm to rely instead on the decomposition, and therefore our work makes this result constructive. The bidimensionality theory, developed in the series =-=[21, 19, 14, 15, 12, 16, 13, 20, 18, 17]-=-, uses the decomposition theorem to develop subexponential fixed-parameter algorithms and PTASs for a broad class of problems in Hminor-free graphs. In [12] a subexponential fixed-parameter algorithm,... |

37 | Bidimensionality: New Connections between FPT Algorithms and PTASs
- Demaine, Hajiaghayi
- 2005
(Show Context)
Citation Context ...tive as mentioned above). We can modify the algorithm to rely instead on the decomposition, and therefore our work makes this result constructive. The bidimensionality theory, developed in the series =-=[21, 19, 14, 15, 12, 16, 13, 20, 18, 17]-=-, uses the decomposition theorem to develop subexponential fixed-parameter algorithms and PTASs for a broad class of problems in Hminor-free graphs. In [12] a subexponential fixed-parameter algorithm,... |

34 | Shallow excluded minors and improved graph decompositions
- Plotkin, Rao, et al.
- 1994
(Show Context)
Citation Context ...lements of the set, the cardinality of the set, or even the order of the largest graph in the set” [28]. Algorithms for H-minor-free graphs for a fixed graph H have been studied extensively; see e.g. =-=[8, 31, 9, 35, 37]-=-. In particular, it is generally believed that several algorithms for planar graphs can be generalized to H-minor-free graphs for any fixed H [31, 35, 37]. The decomposition theorem provides the key i... |

34 | Hadwiger’s conjecture for K6-free graphs
- Robertson, Seymour, et al.
- 1993
(Show Context)
Citation Context ...he conjecture for |V (H)| ≤ 4. The case |V (H)| = 5 is equivalent to the four-color theorem [52], and therefore also true [2, 1, 38]. The case |V (H)| = 6 was proved by Robertson, Seymour, and Thomas =-=[39]-=-, also using the four-color theorem. All cases |V (H)| ≥ 7 remain unsolved. The best general upper bound is that every H-minor-free graph has a vertex coloring with O(|V (H)| � lg |V (H)|) colors, whi... |

28 | Excluding any graph as a minor allows a low tree-width 2-coloring
- DEVOS, DING, et al.
(Show Context)
Citation Context ...d depends on H and k). The proof of this decomposition result is relatively simple, showing the power of our main decomposition result. An existential version of this result was shown by DeVos et al. =-=[22]-=- using a complicated, and not obviously constructive, approach; here we show that a much simpler, and constructive, solution is possible using known results from an earlier paper of Grohe [30]. Even f... |

26 |
Cuts, trees and ℓ1-embeddings of graphs
- Gupta, Newman, et al.
- 1999
(Show Context)
Citation Context ...lements of the set, the cardinality of the set, or even the order of the largest graph in the set” [28]. Algorithms for H-minor-free graphs for a fixed graph H have been studied extensively; see e.g. =-=[8, 31, 9, 35, 37]-=-. In particular, it is generally believed that several algorithms for planar graphs can be generalized to H-minor-free graphs for any fixed H [31, 35, 37]. The decomposition theorem provides the key i... |

25 | Approximation algorithms for classes of graphs excluding single-crossing graphs as minors - Demaine, Hajiaghayi, et al. - 2004 |

24 | Tools for multicoloring with applications to planar graphs and partial k-trees
- Halldórsson, Kortsarz
- 2002
(Show Context)
Citation Context ...to a wide variety of problems to which Baker’s approach applies, such as vertex cover. One particularly interesting application is the wellstudied variation of graph coloring called minimum color sum =-=[4, 26, 34]-=-, where the goal is to find a (vertex or edge) coloring with positive integers with minimum total value. We can use the bounded-treewidth algorithm of Halldórsson and Kortsarz [34], and the merging st... |

23 |
Linear connectivity forces large complete bipartite graph minors
- Böhme, Kawarabayashi, et al.
(Show Context)
Citation Context ...ons of the decomposition theorem include extensions of graph-minor results to countably infinite graphs [23], and the existence of a clique minor whose size is linear in the connectivity of the graph =-=[6]-=-. We believe that our algorithmic decomposition is a useful tool for developing efficient algorithms on H-minor-free graphs. One analogy might be to algorithms for constructing a tree decomposition in... |

22 |
Über eine Eigenschaft der ebene Komplexe
- Wagner
- 1937
(Show Context)
Citation Context ...ee graph has a vertex coloring with |V (H)|−1 colors. Hadwiger [33] posed this problem in 1943, and proved the conjecture for |V (H)| ≤ 4. The case |V (H)| = 5 is equivalent to the four-color theorem =-=[52]-=-, and therefore also true [2, 1, 38]. The case |V (H)| = 6 was proved by Robertson, Seymour, and Thomas [39], also using the four-color theorem. All cases |V (H)| ≥ 7 remain unsolved. The best general... |

18 | Embedding k-outerplanar graphs into ℓ1
- Chekuri, Gupta, et al.
- 2006
(Show Context)
Citation Context |

13 |
Edge-disjoint paths in planar graphs
- Chekuri, Khanna, et al.
- 2004
(Show Context)
Citation Context ...idth at least tw(Ai) − g tw(Ai)/(g + 1) = tw(Ai)/(g + 1), so it has branchwidth Ω(tw(Ai)/(g + 1)) = Ω(tw(Ai)). ✷ 3.7. Half-Integral Versus Fractional Multicommodity Flow Chekuri, Khanna, and Shephard =-=[10]-=- proved that, for planar graphs, the gap between the optimal half-integral multicommodity flow and the optimal fractional multicommodity flow is at most a polylogarithmic factor. Furthermore, they gav... |

12 | The Bidimensional Theory of Bounded-Genus Graphs
- Demaine, Hajiaghayi, et al.
(Show Context)
Citation Context ...tive as mentioned above). We can modify the algorithm to rely instead on the decomposition, and therefore our work makes this result constructive. The bidimensionality theory, developed in the series =-=[21, 19, 14, 15, 12, 16, 13, 20, 18, 17]-=-, uses the decomposition theorem to develop subexponential fixed-parameter algorithms and PTASs for a broad class of problems in Hminor-free graphs. In [12] a subexponential fixed-parameter algorithm,... |

10 |
Efficient Approximation Schemes for Maximization Problems on K3,3-Free Graphs
- Chen
- 1998
(Show Context)
Citation Context ...nded treewidth, for any graph H, and for any ε > 0, there is a polynomial-time (1 + ε)-approximation algorithm for MISP(π) and EMISP(π) on H-minor-free graphs. This result generalizes results of Chen =-=[11]-=- for K3,3minor-free and K5-minor-free graphs, Demaine et al. [19] for single-crossing-minor-free graphs, and Grohe [30] for independent set. Our approach can also obtain a PTAS for maximum P - matchin... |

9 | Diameter and treewidth in minor-closed graph families, revisited, Algorithmica 40
- Demaine, Hajiaghayi
- 2004
(Show Context)
Citation Context |

5 | Dimension reduction in the l1 norm
- Charikar, Sahai
- 2002
(Show Context)
Citation Context |

5 |
algorithms for (k, r)-center in planar graphs and map graphs
- Fixed-parameter
(Show Context)
Citation Context |

5 | Exponential speedup of fixed-parameter algorithms for classes of graphs excluding single-crossing graphs as minors. Algorithmica. To appear. A preliminary version appears
- Demaine, Hajiaghayi, et al.
(Show Context)
Citation Context |

4 |
excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality
- Graphs
- 2005
(Show Context)
Citation Context |

4 | Excluding a countable clique
- Diestel, Thomas
- 1999
(Show Context)
Citation Context ...everal PTASs based on Baker’s approach (from 222O(1/ε) nO(1) to 2O(1/ε) nO(1) ). Other applications of the decomposition theorem include extensions of graph-minor results to countably infinite graphs =-=[23]-=-, and the existence of a clique minor whose size is linear in the connectivity of the graph [6]. We believe that our algorithmic decomposition is a useful tool for developing efficient algorithms on H... |

4 |
The Metamathematics of the Graph Minor Theorem, Logic and Combinatorics
- Friedman, Robertson, et al.
- 1987
(Show Context)
Citation Context ... induction, and gives little insight into the finitely many forbidden minors which are proved to exist. Indeed, there is a mathematical sense in which any proof of this result must be nonconstructive =-=[29]-=-. Essentially the only explicitly algorithmic part of the Graph Minor Theory is a polynomial-time algorithm for testing the existence of fixed minors [44] which, combined with the proof of Wagner’s Co... |

3 |
parameters and local treewidth
- Bidimensional
- 2004
(Show Context)
Citation Context |

1 |
Finding shortest non-contractible and nonzero-homologous cycles for topologically embedded graphs
- CABELLO, MOHAR
- 2004
(Show Context)
Citation Context ...omposition if T = (I, F ) is a path. The pathwidth of a graph G, denoted pw(G), is the minimum width over all possible path decompositions of G. Third, we need a basic notion of embedding; see, e.g., =-=[43, 7]-=-. In this paper, an embedding refers to a 2-cell embedding, i.e., a drawing of the vertices and edges of the graph as points and arcs in a surface such that every face (region outlined by edges) is ho... |