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...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 =-=[17]-=-. Lemma 1 (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 partic6sular, S is a minimal separator if and onl...

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...optimal solutions or lower bounds for diverse optimization problems. (See also [5] for a comprehensive survey.) Treewidth also plays a crucial role in Downey & Fellows parameterized complexity theory =-=[13]-=-. An efficient solution to treewidth is the base for many applications in artificial intelligence, databases and logical-circuit design. See [1] for further references. The minimum fill-in problem has...

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...ue C of a graph G is a subset of V such that all the vertices of C are pairwise adjacent. By ω(G) we denote the maximum clique-size of a graph G. The notion of treewidth is due to Robertson & Seymour =-=[22]-=-. 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 ; (ii...

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... and computational biology. Previous results. Treewidth and minimum fill-in are known to be NP-hard even when the input is restricted to complements of bipartite graphs (so called cobipartite graphs) =-=[2, 25]-=-. Despite of the importance of treewidth almost nothing is known on how to cope with its intractability. It is known that it can be approximated with a factor log OP T [1, 9] and it is an old open que...

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... approximated with a constant factor. Treewidth is known to be fixed parameter tractable, moreover, for a fixed k, the treewidth of size k can be computed in linear time (with a huge hidden constant) =-=[4]-=-. There is a number of algorithms that for a given graph G and integer k, either reports that the treewidth of G is at least k, or produces a tree decomposition of width at most c1 · k in time O(c k 2...

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...n be solved in O ∗ (2 n ) either by using the algorithm of Arnborg et al. [2] or by reformulating them as problems of finding a special vertex ordering and using the technique proposed by Held & Karp =-=[18]-=- for the travelling salesman problem. Our results. In this paper we break the O ∗ (2 n ) barrier by obtaining the first exact algorithm for treewidth of running time O ∗ (c n ) for c < 2. Our algorith...

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... and computational biology. Previous results. Treewidth and minimum fill-in are known to be NP-hard even when the input is restricted to complements of bipartite graphs (so called cobipartite graphs) =-=[2, 25]-=-. Despite of the importance of treewidth almost nothing is known on how to cope with its intractability. It is known that it can be approximated with a factor log OP T [1, 9] and it is an old open que...

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... classical complexity theory, for example why some NP-hard problems can be solved significantly faster than others? For a good overview of the field see the recent survey written by Gerhard Woeginger =-=[24]-=-.sTreewidth is one of the most basic parameters in algorithms and it plays an important role in structural graph theory. It serves as one of the main tools in Robertson & Seymour’s Graph Minors projec...

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...t k, or produces a tree decomposition of width at most c1 · k in time O(c k 2n O(1) ), where c1, c2 are some constants (see e.g. [1]). Fixed parameter algorithms are known for fill-in problem as well =-=[10, 19]-=-. There is no previous work on exact algorithms for treewidth or fill-in and almost nothing was known about it. Both problems can be solved in O ∗ (2 n ) either by using the algorithm of Arnborg et al...

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...le in Downey & Fellows parameterized complexity theory [13]. An efficient solution to treewidth is the base for many applications in artificial intelligence, databases and logical-circuit design. See =-=[1]-=- for further references. The minimum fill-in problem has important applications in sparse matrix computations and computational biology. Previous results. Treewidth and minimum fill-in are known to be...

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...moderate instance sizes. Nice examples of fast exponential algorithms are Eppstein’s graph coloring algorithm with time complexity O ∗ (2.4150 n ) [14] and an O ∗ (1.4802 n ) time algorithm for 3-SAT =-=[12]-=-. (In this paper we use a modified big-Oh notation that suppresses all other (polynomially bounded) terms. For functions f and g we write f(n) = O ∗ (g(n)) if f(n) = O(g(n)poly(|n|)), where poly(|n|) ...

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... project [23]. It is well known that many intractable problems can be solved in polynomial (and very often in linear time) when the input is restricted to graphs of bounded treewidth. In recent years =-=[11]-=- it was shown that the results on graphs of bounded treewidth (branchwidth) are not only of theoretical interest but can successfully be applied to find optimal solutions or lower bounds for diverse o...

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...t k, or produces a tree decomposition of width at most c1 · k in time O(c k 2n O(1) ), where c1, c2 are some constants (see e.g. [1]). Fixed parameter algorithms are known for fill-in problem as well =-=[10, 19]-=-. There is no previous work on exact algorithms for treewidth or fill-in and almost nothing was known about it. Both problems can be solved in O ∗ (2 n ) either by using the algorithm of Arnborg et al...

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...lgorithms for certain NP-hard problems, at least for moderate instance sizes. Nice examples of fast exponential algorithms are Eppstein’s graph coloring algorithm with time complexity O ∗ (2.4150 n ) =-=[14]-=- and an O ∗ (1.4802 n ) time algorithm for 3-SAT [12]. (In this paper we use a modified big-Oh notation that suppresses all other (polynomially bounded) terms. For functions f and g we write f(n) = O ...

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...Treewidth is one of the most basic parameters in algorithms and it plays an important role in structural graph theory. It serves as one of the main tools in Robertson & Seymour’s Graph Minors project =-=[23]-=-. It is well known that many intractable problems can be solved in polynomial (and very often in linear time) when the input is restricted to graphs of bounded treewidth. In recent years [11] it was s...

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...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 =-=[7, 8]-=-. The running time analysis of the algorithm is difficult and is based on a careful counting of potential maximal cliques, i.e. vertex subsets in a graph that can be maximal cliques in some minimal tr...

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... triangulation H of G such that Ω is a maximal clique of H. We denote by ΠG the set of all potential maximal cliques of G. The following result will be used to list all minimal separators. Theorem 2 (=-=[3]-=-). There is an algorithm listing all minimal separators of an input graph G in O(n 3 |∆G|) time. For a set K ⊆ V , a connected component C of G \ K is a full component associated to K if N(C) = K. The...

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...(S, C) is called full if C is a full component associated to S. The graph R(S, C) = GS[S ∪ C] obtained from G[S ∪ C] by completing S into a clique is called the realization of the block B. Theorem 5 (=-=[20]-=-). Let G be a non-complete graph. Then tw(G) = min max S∈∆G C∈C(S) tw(R(S, C)) mfi(G) = min (fill(S) + S∈∆G � mfi(R(S, C))) C∈C(S) where fill(S) is the number of non-edges of G[S]. In the equations of...

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...lled cobipartite graphs) [2, 25]. Despite of the importance of treewidth almost nothing is known on how to cope with its intractability. It is known that it can be approximated with a factor log OP T =-=[1, 9]-=- and it is an old open question if the treewidth can be approximated with a constant factor. Treewidth is known to be fixed parameter tractable, moreover, for a fixed k, the treewidth of size k can be...

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...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 ) =-=[15]-=- 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...

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...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 =-=[7, 8]-=-. The running time analysis of the algorithm is difficult and is based on a careful counting of potential maximal cliques, i.e. vertex subsets in a graph that can be maximal cliques in some minimal tr...

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... n . Our algorithms for treewidth and fill-in can also be applied for solving other problems that can be expressed in terms of minimal triangulations like finding a tree decomposition of minimum cost =-=[6]-=- or computing treewidth of weighted graph. However, there are two ’width’ parameters related to treewidth, namely bandwidth and pathwidth and one parameter called profile, related to minimum fill-in, ...