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## Treewidth: Computational experiments (2001)

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### Citations

2040 |
Network Flows: Theory, Algorithms, and Applications
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ...parating set S with minimum cardinality over all combinations st 62 E. The st-separating set with minimum cardinality can be found eciently using Menger's theorem (see also Ahuja, Magnanti, and Orlin =-=[1]-=-). Theorem 3.7 (Menger [26]) Given a graph G = (V; E) and two distinct non-adjacent vertices s; t 2 V , the minimum number of vertices in an st-separating set is equal to the maximum number of vertex-... |

1524 |
Local computations with probabilities on graphical structures and their applications to expert systems (with discussion),
- Lauritzen, Spiegelhalter
- 1988
(Show Context)
Citation Context ...n the area of expert systems. The currently most ecient algorithm for the inference calculation in probabilistic (or Bayesian) networks builds upon a tree decomposition of a network's moralized graph =-=[20, 24]-=-. All these studies show that (dynamic programming) algorithms based on a path/tree/branch decomposition of the graph can be an alternative for integer programming techniques to solve hard (combinator... |

1419 | Algorithmic Graph Theory and Perfect Graphs - Golumbic - 1980 |

439 |
Graph minors II. Algorithmic aspects of treewidth,
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...thods. For other graphs, however, more sophisticated techniques are necessary. 1 Introduction In the 1980s and early 1990s, Robertson and Seymour dened the graph parameters pathwidths[28], treewidth [=-=29]-=-, and branchwidth [30, 31] as well as the associated graph structures path decomposition, tree decomposition, and branch decomposition. Originally these concepts were introduced in the context of thei... |

386 | Complexity of finding embeddings in a k-tree - Arnborg, Corneil, et al. - 1987 |

378 |
Algorithmic aspects of vertex elimination on graphs
- Rose, Tarjan, et al.
- 1976
(Show Context)
Citation Context ...angulated or return a triangulation of the graph. In the subsequent subsections, three of these algorithms are presented, two variants of the lexicographic breadthsrst search algorithm of Rose et al. =-=[32]-=-, and the maximum cardinality search algorithm of Tarjan and Yannakakis [35]. All three algorithms are based on the characterization of triangulated graphs by a perfect elimination scheme. A vertex v ... |

351 |
Graph minors XIII. The disjoint paths problem
- Robertson, Seymour
- 1995
(Show Context)
Citation Context ...s, however, more sophisticated techniques are necessary. 1 Introduction In the 1980s and early 1990s, Robertson and Seymour dened the graph parameters pathwidths[28], treewidth [29], and branchwidth [=-=30, 31]-=- as well as the associated graph structures path decomposition, tree decomposition, and branch decomposition. Originally these concepts were introduced in the context of their fundamental research on ... |

324 | A partial k-arboretum of graphs with bounded treewidth
- Bodlaender
- 1998
(Show Context)
Citation Context ..., a path decomposition of G is obtained. The minimum width over all possible path decompositions of G is called the pathwidth of G. Many de¯nitions equivalent to treewidth or pathwidth are known, see =-=[7]-=- for an overview. For the heuristics described in the next section, two of these alternative characterizations are exploited. 3 Heuristics In this section, we present four heuristics for computing the... |

321 |
Bayesian updating in causal probabilistic networks by local computations.
- Jensen, Lauritzen, et al.
- 1990
(Show Context)
Citation Context ...n the area of expert systems. The currently most ecient algorithm for the inference calculation in probabilistic (or Bayesian) networks builds upon a tree decomposition of a network's moralized graph =-=[20, 24]-=-. All these studies show that (dynamic programming) algorithms based on a path/tree/branch decomposition of the graph can be an alternative for integer programming techniques to solve hard (combinator... |

312 | Multi-Terminal Network Flows. - Gomory, Hu - 1961 |

306 |
Addendum: simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs,
- Tarjan, Yannakakis
- 1985
(Show Context)
Citation Context ...ons, three of these algorithms are presented, two variants of the lexicographic breadthsrst search algorithm of Rose et al. [32], and the maximum cardinality search algorithm of Tarjan and Yannakakis =-=[35]-=-. All three algorithms are based on the characterization of triangulated graphs by a perfect elimination scheme. A vertex v 2 V is called simplicial if its neighbors N(v) induce a complete subgraph of... |

282 |
Incidence matrices and interval graphs
- Fulkerson, Gross
- 1965
(Show Context)
Citation Context ...; v n g] for all i = 1; : : : ; n. So, G contains a simplicial vertex v 1 , and every time a simplicial vertex v i is removed from the graph, a new simplicial vertex v i+1 exists. Fulkerson and Gross =-=[15]-=- proved that a graph is triangulated if and only if it has a perfect elimination scheme. Given a perfect elimination scheme, the maximal cliques of G are of the form fug [ M(v), where M(v) = fw 2 N(v)... |

268 | A tourist guide through treewidth
- Bodlaender
- 1993
(Show Context)
Citation Context ... problem, the Hamiltonian cycle problem, and the Steiner tree problem. Moreover, several PSPACE-complete problems can be solved in linear time for input restricted to graphs of bounded treewidth (cf. =-=[6]-=-). All these results have been considered primarily of theoretical interest only. In recent years, a number of computational studies has shown that the results can also be applied successfully. Asrst ... |

207 | Graph minors. X. Obstructions to tree-decomposition
- Robertson, Seymour
- 1991
(Show Context)
Citation Context ...s, however, more sophisticated techniques are necessary. 1 Introduction In the 1980s and early 1990s, Robertson and Seymour dened the graph parameters pathwidths[28], treewidth [29], and branchwidth [=-=30, 31]-=- as well as the associated graph structures path decomposition, tree decomposition, and branch decomposition. Originally these concepts were introduced in the context of their fundamental research on ... |

193 |
Efficient algorithms for combinatorial problems on graphs with bounded decomposability – a survey
- Arnborg
- 1985
(Show Context)
Citation Context ...mal cliques of G are of the form fug [ M(v), where M(v) = fw 2 N(v) : 1 (v)s 1 (w)g denotes the set of higher ordered neighbors. The value max v2V jM(v)j is also called the dimension of G. Arnborg [2] proved that the dimension of G equals the treewidth. For a non-triangulated graph G and an elimination scheme , a triangulation G can be constructed by Algorithm 1. Moreover, the algorithm returns... |

168 |
The intersection graphs of subtrees in trees are exactly the chordal graphs
- Gavril
- 1974
(Show Context)
Citation Context ... ¯rst classes of graphs for which it has been proved that they are perfect. The reason for introducing triangulated graphs in this paper is the following property of these graphs: Theorem 3.1 (Gavril =-=[16]-=-) A graph G = (V;E) is triangulated if and only if G can be constructed as the intersection graph of subtrees of trees, i.e., there exists a tree T = (I; F ) such that one can associate a subtree Tv =... |

152 |
Graph minors. I. Excluding a forest.
- Robertson, Seymour
- 1983
(Show Context)
Citation Context ...ied by these methods. For other graphs, however, more sophisticated techniques are necessary. 1 Introduction In the 1980s and early 1990s, Robertson and Seymour dened the graph parameters pathwidths[2=-=8]-=-, treewidth [29], and branchwidth [30, 31] as well as the associated graph structures path decomposition, tree decomposition, and branch decomposition. Originally these concepts were introduced in the... |

151 |
On rigid circuit graphs
- Dirac
- 1961
(Show Context)
Citation Context ...arches for a simplicial vertex, which requires an adjacency-test for all pairs of neighbors of a vertex. More ecient algorithms can be constructed by using another observation. Proposition 3.5 (Dirac =-=[12]-=-) Every triangulated, non-complete graph has two non-adjacent simplicial vertices. This observation (combined with Proposition 3.4) implies that at every step of the above described algorithm we have ... |

134 |
Zur allgemeinen Kurventheorie
- Menger
- 1927
(Show Context)
Citation Context ...m cardinality over all combinations st 62 E. The st-separating set with minimum cardinality can be found eciently using Menger's theorem (see also Ahuja, Magnanti, and Orlin [1]). Theorem 3.7 (Menger =-=[26]-=-) Given a graph G = (V; E) and two distinct non-adjacent vertices s; t 2 V , the minimum number of vertices in an st-separating set is equal to the maximum number of vertex-disjoint paths connecting s... |

125 |
The maximum clique problem
- Pardalos, Xue
- 1994
(Show Context)
Citation Context ...aximum clique number is NP-hard, in general. Nevertheless, several studies have shown that for not too large and not too dense graphs, the maximum clique number can be computed within reasonable time =-=[4, 27]-=-. 4.2 Maximum Minimum Degree Bound A trivial lower bound on the treewidth is given by the minimum degree of the graph: let (T ; X ) be minimal tree decomposition with optimal width k, and let i 2 I be... |

102 |
Preserving order in a forest in less than logarithmic time
- Boas, P
- 1975
(Show Context)
Citation Context ... and Wilf [34] proved that (G) 1 MMD(G). So, a bound that is at least as good as the maximum clique bound and the minimum coloring bound can be found in polynomial time. Using a Van Embe Boas-tree [3=-=6]-=-, the algorithm can be implemented in O(n log log n +m) time. Note that the maximum minimum degree MMD(G) dualizes with another graph theoretic concept called k-degeneracy [25]. A graph is called k-de... |

76 | A sufficiently fast algorithm for finding close to optimal junction trees. - Becket, Geiger - 1996 |

55 | A practical algorithm for finding optimal triangulations - Shoikhet, Geiger - 1997 |

50 |
An inequality for the chromatic number of a graph
- Szekeres, Wilf
- 1968
(Show Context)
Citation Context ... minimum degree vertices of the graph and keep track of the maximum of these minimum degrees, see Algorithm 6. Since the maximum clique of G is a subgraph, MMD(G) !(G) 1. In fact, Szekeres and Wilf [=-=3-=-4] proved that (G) 1 MMD(G). So, a bound that is at least as good as the maximum clique bound and the minimum coloring bound can be found in polynomial time. Using a Van Embe Boas-tree [36], the algo... |

47 | Frequency Assignment - Models and Algorithms
- Koster
- 1999
(Show Context)
Citation Context ...time. 3.2 Minimum Separating Vertex Set Heuristic The fourth heuristic described in this paper was developed in the context of solving frequency assignment problems with a tree decomposition approach =-=[21]-=-. It bases on another characterization of tree decompositions. Every tree decomposition can be transformed to a tree decomposition in which the vertex set associated to an internal node separates the ... |

35 |
Complexity of nding an embedding in k-trees
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context ...tains a large constant with k in the exponent. For small graphs, Shoikhet and Geiger [33] performed various experiments on randomly generated graphs with an adaption of an algorithm of Arnborg et al. =-=[3]-=-. Approximation with an O(log n) approximation ratio can be done in polynomial time (here n denotes the number of vertices in the graph); ¯nding a polynomial time constant approximation ratio algorith... |

32 |
Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring
- Balas, Xue
- 1996
(Show Context)
Citation Context ...aximum clique number is NP-hard, in general. Nevertheless, several studies have shown that for not too large and not too dense graphs, the maximum clique number can be computed within reasonable time =-=[4, 27]-=-. 4.2 Maximum Minimum Degree Bound A trivial lower bound on the treewidth is given by the minimum degree of the graph: let (T ; X ) be minimal tree decomposition with optimal width k, and let i 2 I be... |

24 |
d-degenerate graphs
- Lick
- 1970
(Show Context)
Citation Context ...ng a Van Embe Boas-tree [36], the algorithm can be implemented in O(n log log n +m) time. Note that the maximum minimum degree MMD(G) dualizes with another graph theoretic concept called k-degeneracy =-=[25]-=-. A graph is called k-degenerate if and only if every induced subgraph of G has a vertex of degree at most k. The minimum k for which G is k-degenerate equals MMD(G). 13 Algorithm 6 Maximum Minimum De... |

23 |
Complexity of embeddings in a k-tree
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context ...tains a large constant with k in the exponent. For small graphs, Shoikhet and Geiger [33] performed various experiments on randomly generated graphs with an adaption of an algorithm of Arnborg et al. =-=[3]-=-. Approximation with an O(log n) approximation ratio can be done in polynomial time (here n denotes the number of vertices in the graph);snding a polynomial time constant approximation ratio algorithm... |

23 | Multi-commodity network
- Hu
- 1963
(Show Context)
Citation Context ...btain the minimum separating vertex set. Note that since the graph D 0 is a directed graph, we have to solve O(n 2 ) minimum cut problems. In other words, we cannot use the algorithm of Gomory and Hu =-=[18]-=-, which solves the all pairs minimum cut problem for undirected graphs by solving only O(n) minimum cut problems. 3.2.2 MSVS Heuristic The MSVS heuristic is in fact an improvement heuristic, i.e., it ... |

13 |
Branch decompositions and their applications.
- HICKS
- 2000
(Show Context)
Citation Context ...ally are exponentially in the width. A similar procedure is due for pathwidth and branchwidth. Henceforth, we focus on the computation of the treewidth of graphs from a practical point of view. Hicks =-=[19-=-] recently studied the practical computation of branchwidth for (specic classes of) graphs. The treewidth of a graph is, in general, NP-hard to compute. Much research has been carried out to determine... |

13 |
k-degenerate graphs.
- Lick, White
- 1970
(Show Context)
Citation Context ...ing a Van Embe Boas-tree [36], the algorithm can be implemented in O(n log log n+m) time. Note that the maximum minimum degree MMD(G) dualizes with another graph theoretic concept called k-degeneracy =-=[25]-=-. A graph is called k-degenerate if and only if every induced subgraph of G has a vertex of degree at most k. Hence, the minimum k for which G is kdegenerate equals MMD(G). 13 Algorithm 6 Maximum Mini... |

12 |
Solving frequency assignment problems via tree-decomposition
- Koster, Hoesel, et al.
- 1999
(Show Context)
Citation Context ...o obtain close-to-optimal solutions of the traveling salesman problem. Path decompositions are used by Verweij [37] to solve lifting problems of cycle inequalities for the independent set problem. In =-=[22, 23]-=-, tree decompositions are used to obtain lower bounds and optimal solutions for a special type of frequency assignment problems. Finally, tree decompositions are used to solve problems in the area of ... |

11 |
Selected Applications of Integer Programming: A Computational Study.
- Verweij
- 2000
(Show Context)
Citation Context ...olvability has been undertaken by Cook and Seymour [10]. They use branch decompositions to obtain close-to-optimal solutions of the traveling salesman problem. Path decompositions are used by Verweij =-=[37]-=- to solve lifting problems of cycle inequalities for the independent set problem. In [22, 23], tree decompositions are used to obtain lower bounds and optimal solutions for a special type of frequency... |

10 |
Multi-terminal network °ows
- Gomory, Hu
- 1961
(Show Context)
Citation Context ...e obtain the minimum separating vertex set. Note that since the graph D0 is a directed graph, we have to solve O(n2) minimum cut problems. In other words, we cannot use the algorithm of Gomory and Hu =-=[18]-=-, which solves the all pairs minimum cut problem for undirected graphs by solving only O(n) minimum cut problems. 3.2.2 MSVS Heuristic The MSVS heuristic is in fact an improvement heuristic, i.e., it ... |

7 | Lower bounds for minimum interference frequency assignment problems
- Koster, Hoesel, et al.
- 2001
(Show Context)
Citation Context ...o obtain close-to-optimal solutions of the traveling salesman problem. Path decompositions are used by Verweij [37] to solve lifting problems of cycle inequalities for the independent set problem. In =-=[22, 23]-=-, tree decompositions are used to obtain lower bounds and optimal solutions for a special type of frequency assignment problems. Finally, tree decompositions are used to solve problems in the area of ... |

6 | A Suciently Fast Algorithm for Finding Close to Optimal Junction Trees - Becker, Geiger - 1996 |

5 |
Gaag. Pre-processing for triangulation of probabilistic networks
- Bodlaender, Koster, et al.
- 2001
(Show Context)
Citation Context ...tics of pathogens causing pneumonia 89 154 215 59 170 pathfinder diagnosis of lymphatic disease 109 192 211 14 49 Table 3: Probabilistic network characteristics techniques for computing the treewidth =-=[8]-=-, an additional four instances to conduct our heuristics on are available. The size of these four instances is reported in Table 3 as well. After [8], henceforth we refer to these instances as instanc... |

5 |
A practical algorithm for nding optimal triangulations
- Shokhet, Geiger
- 1997
(Show Context)
Citation Context ...aph has treewidth k have been developed. Their use in practice, however, is very limited since the running time contains a large constant with k in the exponent. For small graphs, Shoikhet and Geiger =-=[33]-=- performed various experiments on randomly generated graphs with an adaption of an algorithm of Arnborg et al. [3]. Approximation with an O(log n) approximation ratio can be done in polynomial time (h... |

4 | Pre-processing for triangulation of probabilistic networks
- Bodlaender, Koster, et al.
- 2001
(Show Context)
Citation Context ...tics of pathogens causing pneumonia 89 154 215 59 170 pathfinder diagnosis of lymphatic disease 109 192 211 14 49 Table 3: Probabilistic network characteristics techniques for computing the treewidth =-=[8]-=-, an additional four instances to conduct our heuristics on are available. The size of these four instances is reported in Table 3 as well. After [8], henceforth we refer to these instances as instanc... |

4 |
An algorithm for the ring-router problem
- Cook, Seymour
- 1994
(Show Context)
Citation Context ...rs, a number of computational studies has shown that the results can also be applied successfully. Asrst attempt to benet from the polynomial time solvability has been undertaken by Cook and Seymour [=-=10]-=-. They use branch decompositions to obtain close-to-optimal solutions of the traveling salesman problem. Path decompositions are used by Verweij [37] to solve lifting problems of cycle inequalities fo... |

4 | Frequency assignment and ramifications of coloring. Discussiones Mathematicae Graph Theory - Eisenblätter, Grötschel, et al. - 2002 |

3 |
A suciently fast algorithm for nding close to optimal clique trees
- Becker, Geiger
- 2001
(Show Context)
Citation Context ...we describe a computational analysis of four heuristics and two lower bounds in this paper. We focus on algorithms that are not exponential in the treewidth (like the algorithm 2 of Becker and Geiger =-=[5]-=-). The heuristics are described in Section 3, whereas Section 4 is devoted to the lower bounds. The computational analysis of the heuristics and lower bounds on several classes of graphs is the topic ... |

2 |
A suÆciently fast algorithm for close to optimal clique trees
- Becker, Geiger
(Show Context)
Citation Context ...we describe a computational analysis of four heuristics and two lower bounds in this paper. We focus on algorithms that are not exponential in the treewidth (like the algorithm 2 of Becker and Geiger =-=[5]-=-). The heuristics are described in Section 3, whereas Section 4 is devoted to the lower bounds. The computational analysis of the heuristics and lower bounds on several classes of graphs is the topic ... |

2 |
project. Publications available at FTP Site: ftp.win.tue.nl, Directory /pub/techreports/CALMA
- CALMA
- 1995
(Show Context)
Citation Context ...probability of the network, or the total interference in the network. For more information on frequency assignment, we refer to [13, 21] or to FAP web [14]. In the context of the EUCLID CALMA project =-=[9]-=-, instances for a variety of frequency assignment problems have been made available. We restrict ourselves to the instances where the objective is to minimize the overall interference in an assignment... |

2 |
FAP web - A website devoted to frequency assignment
- Eisenblatter, Koster
- 2000
(Show Context)
Citation Context ..., the span of the used frequencies, the blocking probability of the network, or the total interference in the network. For more information on frequency assignment, we refer to [13, 21] or to FAP web =-=[14]-=-. In the context of the EUCLID CALMA project [9], instances for a variety of frequency assignment problems have been made available. We restrict ourselves to the instances where the objective is to mi... |

1 |
Frequency assignment and rami cations of coloring. Discussiones Mathematicae Graph Theory, to appear
- Eisenblatter, Grotschel, et al.
- 2001
(Show Context)
Citation Context ...ber of used frequencies, the span of the used frequencies, the blocking probability of the network, or the total interference in the network. For more information on frequency assignment, we refer to =-=[13, 21]-=- or to FAP web [14]. In the context of the EUCLID CALMA project [9], instances for a variety of frequency assignment problems have been made available. We restrict ourselves to the instances where the... |

1 |
A practical algorithm for optimal triangulations
- Shoikhet, Geiger
- 1997
(Show Context)
Citation Context ...aph has treewidth k have been developed. Their use in practice, however, is very limited since the running time contains a large constant with k in the exponent. For small graphs, Shoikhet and Geiger =-=[33]-=- performed various experiments on randomly generated graphs with an adaption of an algorithm of Arnborg et al. [3]. Approximation with an O(log n) approximation ratio can be done in polynomial time (h... |

1 | The second DIMACS implementation challenge: NP-Hard Problems: Maximum Clique, Graph Coloring, and Satisfiability. See http://dimacs.rutgers.edu/Challenges/ or http://mat.gsia.cmu.edu/COLOR/instances.html - Dirac - 1992 |

1 |
Frequency assignment and rami¯- cations of coloring. Discussiones Mathematicae Graph Theory, to appear
- EisenblÄatter, GrÄotschel, et al.
- 2001
(Show Context)
Citation Context ...ber of used frequencies, the span of the used frequencies, the blocking probability of the network, or the total interference in the network. For more information on frequency assignment, we refer to =-=[13, 21]-=- or to FAP web [14]. In the context of the EUCLID CALMA project [9], instances for a variety of frequency assignment problems have been made available. We restrict ourselves to the instances where the... |