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...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-...

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...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...

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...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...

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...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 ...

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...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 ...

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..., 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...

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...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...

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...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...

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...; 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)...

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... 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 ...

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...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 ...

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...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...

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... ¯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 =...

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...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...

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...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 ...

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...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...

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...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...

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... 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...

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... 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...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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...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 ...

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...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...

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...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...

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...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 ...

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...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...

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...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 ...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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...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 ...

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...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 ...

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...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...

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..., 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...

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...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...

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...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...

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...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...