## Exact algorithms for treewidth and minimum fill-in (2004)

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Venue: | In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004). Lecture Notes in Comput. Sci |

Citations: | 24 - 14 self |

### BibTeX

@INPROCEEDINGS{Fomin04exactalgorithms,

author = {Fedor V. Fomin and Ioan Todinca and Dieter Kratsch and Yngve Villanger},

title = {Exact algorithms for treewidth and minimum fill-in},

booktitle = {In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004). Lecture Notes in Comput. Sci},

year = {2004},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be reduced to O(1.4142 n).

### Citations

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Citation Context ...ere the best upper bound we were able to find. Let S be a separator in a graph G = (V, E). For x ∈ V \ S, we denote by Cx(S) the component of G \ S containing x. The following lemma is an exercise in =-=[27]-=-. Lemma 9 (Folklore). A set S of vertices of G is a minimal a, b-separator if and only if a and b are in different full components associated to S. In particular, S is a minimal separator if and only ... |

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Citation Context ...ortant role in structural Graph Theory. It serves as one of the main tools in Robertson and Seymour’s Graph Minors project [39]. Treewidth also plays a crucial role in parameterized complexity theory =-=[19]-=-. The minimum fill-in problem (also known as minimum chordal graph completion) has important applications in sparse matrix computations and computational biology. The problems to compute the treewidth... |

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Citation Context ...hat all the vertices of C are pairwise adjacent. By ω(G) we denote the maximum clique-size of a graph G. Treewidth and minimum fill-in of graphs. The notion of treewidth is due to Robertson & Seymour =-=[38]-=-. A tree decomposition of a graph G = (V, E), denoted by T D(G), is a pair (X, T ) in which T = (VT , ET ) is a tree and X = {Xi | i ∈ VT } is a family of subsets of V such that: (i) ⋃ i∈VT Xi = V ; (... |

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Citation Context ...ology. The problems to compute the treewidth and minimum fill-in of a graph are known to be NP-hard even when the input is restricted to complements of bipartite graphs (so called cobipartite graphs) =-=[2, 48]-=-. Despite of the importance of treewidth almost nothing is known on how to cope with its intractability. For a long time the best known approximation algorithm for treewidth had a factor log OP T [1, ... |

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Citation Context ...er tractable. Moreover, for any fixed k, there is a linear time algorithm to compute the treewidth of graphs of treewidth at most k (unfortunately there is a huge hidden constant in the running time) =-=[6]-=-. There is a number of algorithms that for a given graph G and integer k, either report that the treewidth of G is at least k, or produce a tree decomposition of width at most c1·k in time c2 k ·n O(1... |

183 |
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Citation Context ...iology. The problems to compute the treewidth and minimum fill-in of a graph are known to be NPhard even when the input is restricted to complements of bipartite graphs (so called cobipartite graphs) =-=[2, 48]-=-. Despite of the importance of treewidth almost nothing is known on how to cope with its intractability. For a long time the best known approximation algorithm for treewidth had a factor log OP T [1, ... |

163 | Graph Minors X. Obstructions to Treedecomposition
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Citation Context ... one of the most basic parameters in Graph Algorithms [7] and it plays an important role in structural Graph Theory. It serves as one of the main tools in Robertson and Seymour’s Graph Minors project =-=[39]-=-. Treewidth also plays a crucial role in parameterized complexity theory [19]. The minimum fill-in problem (also known as minimum chordal graph completion) has important applications in sparse matrix ... |

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Citation Context ...inimal separators, potential maximal clique, minimal triangulation 1 Introduction Exact exponential algorithms. The interest in exact (fast) exponential algorithms dates back to Held and Karp’s paper =-=[28]-=- on the travelling salesman problem in the early sixties. To mention just a few examples: a O(1.4422 n ) 1 time algorithm for Knapsack (Horowitz and Sahni [29]); O(1.2600 n ) and O(1.2109 n ) time alg... |

123 | Which problems have strongly exponential complexity
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Citation Context ...days, it is common believe that NP-hard problems cannot be solved in polynomial time. For a number of NP-hard problems, we even have strong evidence that they cannot be solved in sub-exponential time =-=[30]-=-. In order to obtain exact solutions to these problems, the only hope is to design exact algorithms with good exponential running times. The last years have seen an emerging interest in attacking this... |

113 | Exact algorithms for NP-hard problems: a survey
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Citation Context ...r can enlarge the mentioned size only by a constant additive factor. For overviews and introductions to the field see the recent surveys by Fomin et al. [24], Iwama [31], Schöning [41], and Woeginger =-=[46, 47]-=-. Treewidth and minimum fill-in. Treewidth is one of the most basic parameters in Graph Algorithms [7] and it plays an important role in structural Graph Theory. It serves as one of the main tools in ... |

95 |
Algorithms for maximum independent sets
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Citation Context ...ion just a few examples: a O(1.4422 n ) 1 time algorithm for Knapsack (Horowitz and Sahni [29]); O(1.2600 n ) and O(1.2109 n ) time algorithms for Independent Set (Tarjan and Trojanowski [43], Robson =-=[40]-=-); 3-Coloring in time O(1.4422 n ) (Lawler [35]); 3-SAT in time O(1.6181 n ) (Monien and Speckenmeyer [36]). ∗ A preliminary version of these results appeared in [25] and [44] † Supported by The Auror... |

89 | Efficient approximation for triangulation of minimum treewidth
- Amir
- 2001
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Citation Context ... 48]. Despite of the importance of treewidth almost nothing is known on how to cope with its intractability. For a long time the best known approximation algorithm for treewidth had a factor log OP T =-=[1, 11]-=- (see also [10]). Recently, Feige et al. [21] obtained factor √ log OP T approximation algorithm for treewidth. Furthermore it is an old open question whether the treewidth can be approximated within ... |

88 |
Call routing and the ratcatcher
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- 1994
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Citation Context ...restricted to planar graphs is a long standing open problem in Graph Algorithms. The treewidth of planar graphs can be approximated within a constant factor of 1.5. More precisely, Seymour and Thomas =-=[42]-=- gave a polynomial algorithm for computing the branchwidth of planar graphs, and the latter parameter differs by at most a factor of 1.5 from the treewidth. In the case of planar graphs with n vertice... |

79 |
Computing partitions with applications to the knapsack problem
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- 1974
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Citation Context ...thms dates back to Held and Karp’s paper [28] on the travelling salesman problem in the early sixties. To mention just a few examples: a O(1.4422 n ) 1 time algorithm for Knapsack (Horowitz and Sahni =-=[29]-=-); O(1.2600 n ) and O(1.2109 n ) time algorithms for Independent Set (Tarjan and Trojanowski [43], Robson [40]); 3-Coloring in time O(1.4422 n ) (Lawler [35]); 3-SAT in time O(1.6181 n ) (Monien and S... |

79 |
Finding a maximum independent set
- Tarjan, Trojanowski
- 1986
(Show Context)
Citation Context ...ties. To mention just a few examples: a O(1.4422 n ) 1 time algorithm for Knapsack (Horowitz and Sahni [29]); O(1.2600 n ) and O(1.2109 n ) time algorithms for Independent Set (Tarjan and Trojanowski =-=[43]-=-, Robson [40]); 3Coloring in time O(1.4422 n ) (Lawler [35]); 3-SAT in time O(1.6181 n ) (Monien and Speckenmeyer [36]). Nowadays, it is common believe that NP-hard problems cannot be solved in polyno... |

69 |
Solving satisfiability in less than 2 n steps
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- 1985
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Citation Context ...n ) and O(1.2109 n ) time algorithms for Independent Set (Tarjan and Trojanowski [43], Robson [40]); 3-Coloring in time O(1.4422 n ) (Lawler [35]); 3-SAT in time O(1.6181 n ) (Monien and Speckenmeyer =-=[36]-=-). ∗ A preliminary version of these results appeared in [25] and [44] † Supported by The Aurora Programme Collaboration Research Project between Norway and France. ‡ Fedor Fomin acknowledges support o... |

65 | Improved Approximation Algorithms for Minimum-Weight Vertex Separators
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- 2005
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Citation Context ...ost nothing is known on how to cope with its intractability. For a long time the best known approximation algorithm for treewidth had a factor log OP T [1, 11] (see also [10]). Recently, Feige et al. =-=[21]-=- obtained factor √ log OP T approximation algorithm for treewidth. Furthermore it is an old open question whether the treewidth can be approximated within a constant factor. 2Treewidth is known to be... |

64 |
Fixed-parameter tractability of graph modification problems for hereditary properties
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- 1996
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Citation Context ... produce a tree decomposition of width at most c1·k in time c2 k ·n O(1) , where c1, c2 are some constants (see e.g. [1]). Fixed parameter algorithms are known for the minimum fill-in problem as well =-=[16, 32]-=-. There exists an exact O(2.9512 n ) time algorithm that computes the treewidth of a graph in polynomial space [9]. We are not aware about any previous work on exact algorithms for the treewidth or mi... |

58 |
A note on the complexity of the chromatic number problem
- Lawler
- 1976
(Show Context)
Citation Context ...algorithm for Knapsack (Horowitz and Sahni [29]); O(1.2600 n ) and O(1.2109 n ) time algorithms for Independent Set (Tarjan and Trojanowski [43], Robson [40]); 3-Coloring in time O(1.4422 n ) (Lawler =-=[35]-=-); 3-SAT in time O(1.6181 n ) (Monien and Speckenmeyer [36]). ∗ A preliminary version of these results appeared in [25] and [44] † Supported by The Aurora Programme Collaboration Research Project betw... |

55 | Approximating treewidth, pathwidth, frontsize, and shortest elimination tree, J.Algorithms
- Bodlaender, Gilbert, et al.
- 1995
(Show Context)
Citation Context ...he importance of treewidth almost nothing is known on how to cope with its intractability. For a long time the best known approximation algorithm for treewidth had a factor log OP T [1, 11] (see also =-=[10]-=-). Recently, Feige et al. [21] obtained factor √ log OP T approximation algorithm for treewidth. Furthermore it is an old open question whether the treewidth can be approximated within a constant fact... |

54 | Asteroidal triple-free graphs
- Corneil, Olariu, et al.
- 1997
(Show Context)
Citation Context ...oiding the neighborhood of the third vertex. Graphs without asteroidal triples are called AT-free. Corneil, Olariu & Stewart studied structural properties of AT-free graphs in their fundamental paper =-=[17]-=-. Among others they showed that every connected AT-free graph has a dominating pair, where two vertices x and y of G form a dominating pair (DP for short) if the vertex set of each x, y-path is a domi... |

47 | Measure and conquer: Domination - A case study
- Fomin, Grandoni, et al.
(Show Context)
Citation Context ...); an O(1.7325 n ) time algorithm for Max-Cut (Williams [45]); an algorithm for 3-SAT in time O(1.4726 n ) (Brueggemann and Kern [15]); an O(1.5129 n ) time algorithm for Dominating Set (Fomin et al. =-=[23]-=-). There can be several explanations why now the algorithmic community witnesses the revival of the interest in fast exponential algorithms: • The design and analysis of exact algorithms leads to a be... |

33 |
3-coloring in time O(1.3289
- Beigel, Eppstein
(Show Context)
Citation Context ...combinatorial problems: There are for example an O ∗ (2 n ) time algorithm for Coloring (Björklund, Husfeldt, and Koivisto [5, 34]); an O(1.3289 n ) time algorithm for 3-Coloring (Beigel and Eppstein =-=[3]-=-); an O(1.7325 n ) time algorithm for Max-Cut (Williams [45]); an algorithm for 3-SAT in time O(1.4726 n ) (Brueggemann and Kern [15]); an O(1.5129 n ) time algorithm for Dominating Set (Fomin et al. ... |

33 | Generating all the minimal separators of a graph
- Berry, Bordat, et al.
- 1999
(Show Context)
Citation Context ... n and |ΠG| ≤ 2 n for every graph G on n vertices, and no better upper bounds had been known prior to our work. The following result will be used to list all minimal separators of a graph. Theorem 2 (=-=[4]-=-). There is an algorithm listing all minimal separators of an input graph G in O(n 3 |∆G|) time. There is a very useful relationship between the minimal separators of a graph and its minimal triangula... |

33 |
Characterizations and algorithmic applications of chordal graph embeddings
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- 1997
(Show Context)
Citation Context ...parators S and T of a graph G are said to be crossing if S is a minimal u, v-separator for a pair of vertices u, v ∈ T , in which case T is a minimal x, y-separator for a pair x, y ∈ S. (See [33] and =-=[37]-=- for a full proof.) Theorem 3 ([37]). The graph H is a minimal triangulation of the graph G if and only if there is a maximal set of pairwise non-crossing minimal separators {S1, S2, . . . , Sp} of G ... |

30 | A new algorithm for optimal constraint satisfaction and its implications
- Williams
- 2004
(Show Context)
Citation Context ... time algorithm for Coloring (Björklund, Husfeldt, and Koivisto [5, 34]); an O(1.3289 n ) time algorithm for 3-Coloring (Beigel and Eppstein [3]); an O(1.7325 n ) time algorithm for Max-Cut (Williams =-=[45]-=-); an algorithm for 3-SAT in time O(1.4726 n ) (Brueggemann and Kern [15]); an O(1.5129 n ) time algorithm for Dominating Set (Fomin et al. [23]). There can be several explanations why now the algorit... |

28 | Treewidth and minimum fill-in: Grouping the minimal separators
- Bouchitté, Todinca
(Show Context)
Citation Context ...ithm computing the treewidth and minimum fill-in of a graph on n vertices. The algorithm can be regarded as dynamic programming across partial solutions and is based on results of Bouchitté & Todinca =-=[13, 14]-=-. The analysis of the running time is difficult and is based on combinatorial properties of special structures in a graph, namely, potential maximal cliques, i.e. vertex subsets in a graph that can be... |

27 |
On treewidth and minimum fill-in of asteroidal triple-free graphs
- Kloks, Kratsch, et al.
- 1997
(Show Context)
Citation Context ...inimal separators S and T of a graph G are said to be crossing if S is a minimal u, v-separator for a pair of vertices u, v ∈ T , in which case T is a minimal x, y-separator for a pair x, y ∈ S. (See =-=[33]-=- and [37] for a full proof.) Theorem 3 ([37]). The graph H is a minimal triangulation of the graph G if and only if there is a maximal set of pairwise non-crossing minimal separators {S1, S2, . . . , ... |

24 |
Inclusion-exclusion algorithms for counting set partitions
- BJÖRKLUND, HUSFELDT
(Show Context)
Citation Context ...years have seen an emerging interest in attacking this question for concrete combinatorial problems: There are for example an O ∗ (2 n ) time algorithm for Coloring (Björklund, Husfeldt, and Koivisto =-=[5, 34]-=-); an O(1.3289 n ) time algorithm for 3-Coloring (Beigel and Eppstein [3]); an O(1.7325 n ) time algorithm for Max-Cut (Williams [45]); an algorithm for 3-SAT in time O(1.4726 n ) (Brueggemann and Ker... |

22 |
On treewidth approximations
- Bouchitté, Kratsch, et al.
- 2004
(Show Context)
Citation Context ... 48]. Despite of the importance of treewidth almost nothing is known on how to cope with its intractability. For a long time the best known approximation algorithm for treewidth had a factor log OP T =-=[1, 11]-=- (see also [10]). Recently, Feige et al. [21] obtained factor √ log OP T approximation algorithm for treewidth. Furthermore it is an old open question whether the treewidth can be approximated within ... |

22 | Exact (exponential) algorithms for treewidth and minimum fill-in
- Fomin, Kratsch, et al.
(Show Context)
Citation Context ...rjan and Trojanowski [43], Robson [40]); 3-Coloring in time O(1.4422 n ) (Lawler [35]); 3-SAT in time O(1.6181 n ) (Monien and Speckenmeyer [36]). ∗ A preliminary version of these results appeared in =-=[25]-=- and [44] † Supported by The Aurora Programme Collaboration Research Project between Norway and France. ‡ Fedor Fomin acknowledges support of the Norwegian Research Council. 1 n O(1) n Because c · n =... |

19 | An O(2 n ) algorithm for graph coloring and other partitioning problems via inclusionexclusion
- KOIVISTO
(Show Context)
Citation Context ...years have seen an emerging interest in attacking this question for concrete combinatorial problems: There are for example an O ∗ (2 n ) time algorithm for Coloring (Björklund, Husfeldt, and Koivisto =-=[5, 34]-=-); an O(1.3289 n ) time algorithm for 3-Coloring (Beigel and Eppstein [3]); an O(1.7325 n ) time algorithm for Max-Cut (Williams [45]); an algorithm for 3-SAT in time O(1.4726 n ) (Brueggemann and Ker... |

19 |
Algorithmics in exponential time
- Schöning
- 2005
(Show Context)
Citation Context ... on a faster computer can enlarge the mentioned size only by a constant additive factor. For overviews and introductions to the field see the recent surveys by Fomin et al. [24], Iwama [31], Schöning =-=[41]-=-, and Woeginger [46, 47]. Treewidth and minimum fill-in. Treewidth is one of the most basic parameters in Graph Algorithms [7] and it plays an important role in structural Graph Theory. It serves as o... |

18 |
An improved deterministic local search algorithm for 3-SAT
- Brueggemann, Kern
- 2004
(Show Context)
Citation Context ... an O(1.3289 n ) time algorithm for 3-Coloring (Beigel and Eppstein [3]); an O(1.7325 n ) time algorithm for Max-Cut (Williams [45]); an algorithm for 3-SAT in time O(1.4726 n ) (Brueggemann and Kern =-=[15]-=-); an O(1.5129 n ) time algorithm for Dominating Set (Fomin et al. [23]). There can be several explanations why now the algorithmic community witnesses the revival of the interest in fast exponential ... |

17 | Nondeterministic Graph Searching: From Pathwidth to Treewidth
- Fomin, Fraigniaud, et al.
- 2005
(Show Context)
Citation Context ... similarly fill-in) in time O ∗ (2 n ); • Both problems can be solved by making use of the game theoretic approach, by finding a specific path in the graph of possible states of a Cop and Robber game =-=[22]-=-.EXACT ALGORITHMS FOR TREEWIDTH 3 However it is not clear whether any of the mentioned approaches can bring us to a O(c n ) time algorithm for some c < 2. Prior to our work no exact algorithm computi... |

16 | On Exact Algorithms for Treewidth
- Bodlaender, Fomin, et al.
(Show Context)
Citation Context ...]). Fixed parameter algorithms are known for the minimum fill-in problem as well [16, 32]. There exists an exact O(2.9512 n ) time algorithm that computes the treewidth of a graph in polynomial space =-=[9]-=-. We are not aware about any previous work on exact algorithms for the treewidth or minimum fill-in problem that solves the problem in O(c n ) time where c < 2. There are three relatively simple appro... |

14 |
Coping with the NP-hardness of the graph bandwidth problem
- Feige
- 2000
(Show Context)
Citation Context ...s related to treewidth, namely bandwidth and pathwidth and one parameter called profile, related to minimum fill-in, that do not fit into this framework. Bandwidth can be computed in time O ∗ (10 n ) =-=[20]-=- and reducing Feige’s bounds is a challenging problem. Pathwidth (and profile) can be expressed as vertex ordering problems and thus solved in O ∗ (2 n ) time by applying a dynamic programming approac... |

13 |
Tarjan, Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs
- Kaplan, Shamir, et al.
- 1906
(Show Context)
Citation Context ... produce a tree decomposition of width at most c1·k in time c2 k ·n O(1) , where c1, c2 are some constants (see e.g. [1]). Fixed parameter algorithms are known for the minimum fill-in problem as well =-=[16, 32]-=-. There exists an exact O(2.9512 n ) time algorithm that computes the treewidth of a graph in polynomial space [9]. We are not aware about any previous work on exact algorithms for the treewidth or mi... |

12 |
Thilikos. New upper bounds on the decomposability of planar graphs
- Fomin, M
- 2005
(Show Context)
Citation Context ...anchwidth of planar graphs, and the latter parameter differs by at most a factor of 1.5 from the treewidth. In the case of planar graphs with n vertices, the treewidth is at most O( √ n). Theorem 42 (=-=[26]-=-). For any planar graph G on n vertices, tw(G) ≤ 3.182 √ n + O(1). Also given a graph G and a number k, one can decide if tw(G) ≤ k in O ∗ (n k ) time, either using the algorithm of Arnborg et al. [2]... |

12 |
Worst-case upper bounds for kSAT
- Iwama
- 2004
(Show Context)
Citation Context ...ntial algorithm on a faster computer can enlarge the mentioned size only by a constant additive factor. For overviews and introductions to the field see the recent surveys by Fomin et al. [24], Iwama =-=[31]-=-, Schöning [41], and Woeginger [46, 47]. Treewidth and minimum fill-in. Treewidth is one of the most basic parameters in Graph Algorithms [7] and it plays an important role in structural Graph Theory.... |

9 |
Chordal Embeddings of Planar Graphs
- Bouchitté, Mazoit, et al.
- 2003
(Show Context)
Citation Context ...y the treewidth of planar graphs can be computed in time O ∗ (n 3.182√ n ) = 2 O(√ n log n) . 22Unfortunately, although the structure of potential maximal cliques in planar graphs is very particular =-=[12]-=-, our approach can not be used for obtaining algorithms of running time 2 O(√ n) for planar treewidth. This is because the number of ’large’ potential maximal cliques in planar graphs can be ’large’. ... |

9 |
Improved exponential-time algorithms for treewidth and minimum fill-in
- Villanger
(Show Context)
Citation Context ...Trojanowski [43], Robson [40]); 3-Coloring in time O(1.4422 n ) (Lawler [35]); 3-SAT in time O(1.6181 n ) (Monien and Speckenmeyer [36]). ∗ A preliminary version of these results appeared in [25] and =-=[44]-=- † Supported by The Aurora Programme Collaboration Research Project between Norway and France. ‡ Fedor Fomin acknowledges support of the Norwegian Research Council. 1 n O(1) n Because c · n = O((c + ɛ... |

6 | Tree decompositions with small cost
- Bodlaender, Fomin
- 2002
(Show Context)
Citation Context ...Our algorithms for treewidth and minimum fill-in can also be used for solving other problems that can be expressed in terms of minimal triangulations like finding a tree decomposition of minimum cost =-=[8]-=- or computing treewidth of weighted graphs. However, there are two ’width’ parameters related to treewidth, namely bandwidth and pathwidth and one parameter called profile, related to minimum fill-in,... |

5 |
and time complexity of exact algorithms: Some open problems
- Space
- 2004
(Show Context)
Citation Context ...r can enlarge the mentioned size only by a constant additive factor. For overviews and introductions to the field see the recent surveys by Fomin et al. [24], Iwama [31], Schöning [41], and Woeginger =-=[46, 47]-=-. Treewidth and minimum fill-in. Treewidth is one of the most basic parameters in Graph Algorithms [7] and it plays an important role in structural Graph Theory. It serves as one of the main tools in ... |

1 |
time algorithms for dominating pairs in asteroidal triple-free graphs
- Linear
- 1999
(Show Context)
Citation Context ...n/2 +3 relies on properties of 2LexBFS, i.e. a combination of two runs of lexicographic breadthfirst-search (also called 2-sweep LexBFS), on AT-free graphs established by Corneil, Olariu & Stewart in =-=[18]-=-. Definition 37. A vertex ordering xn, xn−1, . . . , x1 is said to be a 2LexBFS ordering of G if some 2LexBFS(G) returns the vertices in this order (starting with xn) during the second sweep of LexBFS... |