S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math., 23:11--24, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that can be stated as a logic formula of a certain type can be solved in linear time on graphs with bounded tree width [1, 6] Many well known graph problems can be formulates as such formulas: maximum clique, maximum independent set, chromatic number and so on. However, mccp and cpp cannot [2, 4]. It is still open whether there is a linear time algorithm for cpp (and mccp) on mixed graphs with bounded tree width. 3 The minimum cycle cover problem The algorithm for mccp follows the same lines of the algorithm described above. The di erence lies in how we de ne a partial solution for ....

S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete App. Math., 23 (1989), 11-24.

....The treewidth of a graph G (tw(G) is the minimum width over all possible tree decompositions of G. For many NP complete problems, we can derive polynomial time algorithms restricted to graphs of bounded treewidth using a general dynamic programming approach similar to that on trees [4]. However, still the class of graphs of bounded treewidth is of limited size; we would like to solve NP complete problems for wider classes of graphs. As mentioned before, Baker [5] developed several approximation algorithms to solve NP complete problems for planar graphs. To extend these ....

....the number of G i s. ut For example, for G be a non negative vertex weighted apex minor free graph, the maximum independent set problem admits a polynomial time approximation scheme of ration 1 1=k with running time O(k2 k4 11=3(ck d) jV j) see the dynamic programming for this problem in [4]. In fact, the proof of Theorem 7 can be easily generalized for many other problems. For the sake of similarity of the proof, we only mention the general theorem. Theorem 8. Given an H minor free graph G, where H is an apex graph, there are PTASs with approximation ratio 1 1=k (or 1 2=k) ....

Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11-24, 1989.

....The treewidth of a graph G (tw(G) is the minimum width over all possible tree decompositions of G. For many NP complete problems, we can derive polynomial time algorithms restricted to graphs of bounded treewidth using a general dynamic programming approach similar to that on trees [4]. However, still the class of graphs of bounded treewidth is of limited size; we would like to solve NP complete problems for wider classes of graphs. As mentioned before, Baker [5] developed several approximation algorithms to solve NP complete problems for planar graphs. To extend these ....

....number of G i s. For example, for G be a non negative vertex weighted apex minor free graph, the maximum independent set problem admits a polynomial time approximation scheme of ration 1 1 k with running time O(k2 k4 11 3(ck d) V ) see the dynamic programming for this problem in [4]. In fact, the proof of Theorem 7 can be easily generalized for many other problems. For the sake of similarity of the proof, we only mention the general theorem. Theorem 8. Given an H minor free graph G, where H is an apex graph, there are PTASs with approximation ratio 1 1 k (or 1 2 k) ....

Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11--24, 1989.

....Experimental results show that these reduction rules can signi cantly reduce the problem size for several instances of real life probabilistic networks. 1 Introduction For many graph problems, it is useful and important to nd a tree decomposition of minimal width (called treewidth) [4, 11, 13]. Often these problems can be solved in linear or polynomial time when a tree decomposition of bounded width is known. The problem of nding a tree decomposition with minimum width is NP hard [1] and approximating treewidth is also NP hard [10] In [6] it was shown that preprocessing techniques ....

....4, we consider several special cases of the Contraction Reduction rule that can be identi ed in polynomial time. Unweighted variants of these instances, specially the Almost Simplicial, Buddy and (Extended) Cube rules have been introduced and proven safe for unweighted treewidth previously in [4, 6]. We thus have generalised these rules to the weighted case. In addition, we obtain new rules which are safe for weighted treewidth. The Buddies rule generalises the Buddy rule and is an example of such a new rule. As for the unweighted case, the contraction reduction rule bases on a variable low ....

S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math., 23:11-24, 1989.

....are restricted to graphs of bounded treewidth. There are a few techniques for obtaining such algorithms. The main technique is called computing tables of characterizations of partial solutions. This technique is a general dynamic programming approach, first introduced by Arnborg and Proskurowski [AP89] Bodlaender [Bod97] gave a better presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [ACPS93,BdF96] and describing the problems in certain types of logic [ALS88,Cou90] 2.3 Fixed parameter tractability Developing ....

Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11--24, 1989.

....a polynomial time approximation scheme of ratio 1 1=k with running time O(k 4 for G K 5 minor free. 19 Proof. Using dynamic programming on a tree decomposition, this problem can be solved in O(4 n) time, over each n vertex partial w tree whose tree decomposition is given [AP89]. Thus the result follows from Theorem 10 for Time (w; n) O(4 n) Below we give examples that show how our result can be applied to NP minimization problems, e.g. the minimum vertex cover problem and the minimum dominating set problem. The ideas of the proofs of Theorems 11 and 12 ....

Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11-24, 1989.

....when they are restricted to graphs of bounded treewidth. There are a few techniques for obtaining such algorithms. The main technique is called computing tables of characterizations of partial solutions. This technique is a dynamic programming approach, first introduced by Arnborg and Proskurowski [AP89]. This technique also appeared in a paper written by Bern et al. BLW87] Bodlaender [Bod97] described a better presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [ACPS93,BdF96] and describing the problems in logic ....

....with running time O(k Delta 4 Delta n) O(k Delta 4 Delta n k Delta n ) for maximum independent set. Proof. Using dynamic programming on a tree decomposition, this problem can be solved in O(4 Delta n) time, over each n vertex partial w tree whose tree decomposition is given [AP89]. Thus Time (w; n) O(4 Delta n) and the result follows from Theorem 12. ut Below we give examples that show how our result can be applied to NP minimization problems, e.g. the minimum vertex cover problem and the minimum dominating set problem. The ideas of the proofs of Theorems 13 and 14 ....

Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11--24, 1989.

....a k tree by adding a new vertex v adjacent to each vertex of a clique C with at most k vertices is also a k tree. This definition of a k tree is by Rautenbach and Reed [27] The following more restrictive definition of a k tree, which we call strict , was introduced by Arnborg and Proskurowski [1] and is more often used in the literature. A k clique is a strict k tree, and the graph obtained from a strict k tree by adding a new vertex v adjacent to each vertex of a k clique is also a strict k tree. Obviously the strict k trees are a proper sub class of the k trees. A subgraph of a k tree ....

S. ARNBORG AND A. PROSKUROWSKI, Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11--24, 1989.

.... with treewidth at most k is exactly the class of partial k trees (see e.g. 21] 16] There exist linear time algorithms for many NP complete problems, when restricted to the class of partial k trees for some constant k and when a tree decomposition with bounded width is given (see e.g. 1] [5], 7] 3] and [21] Determining whether the treewidth or pathwidth of a given graph is at most a given integer k is NP complete ( 2] In view of this the results of Robertson and Seymour on minor closed classes of graph are of great interest. Definition 2.6 An elementary contraction of a graph ....

S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math., 23, 11 - 24, 1989.

....inside and outside graph theory. One reason for this interest is that many graph problems, including many well known and important problems, become polynomial time, and usually even linear time solvable (and become member of NC) when restricted to a class of graphs with bounded tree or pathwidth [1, 3, 4, 5, 7, 27]. In general, such algorithms need to have a tree decomposition or path decomposition of suitable width given together with the input graph. Hence, an important problem is to find treedecompositions (or path decompositions) of minimum width. When the desired width of the tree decomposition is ....

S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Applied Mathematics 23, 11 \Gamma 24, 1989. 20

....are restricted to graphs of bounded treewidth. There are a few techniques for obtaining such algorithms. The main technique is called computing tables of characterizations of partial solutions. This technique is a general dynamic programming approach, rst introduced by Arnborg and Proskurowski [AP89] Bodlaender [Bod97] described a better presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [ACPS93,BdF96] and describing the problems in certain types of logic [ALS88,Cou90] 2.3 Fixed parameter tractability ....

Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11-24, 1989.

....to several applications in graph theory and other areas. One reason for this interest is that many well known and important graph problems become polynomial time, and usually even linear time solvable (and become member of NC) when restricted to a class of graphs with bounded tree or pathwidth [1, 3, 4, 5, 7]. In general, such algorithms need to have a tree decomposition or path decomposition of suitable width given together with the input graph. Hence, an important problem is to nd tree decompositions (or path decompositions) of minimum width. When the desired width of the tree decomposition is ....

S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Applied Mathematics 23, 11 24, 1989.

....better understood. The associated complexity analysis and the connection to graph parameters are also made explicit. Task specific properties are also studied (e. g, irrelevant buckets in belief updating) The work we show here also fits into the framework developed by Arnborg and Proskourowski [2, 1]. They present table based reductions for various NPhard graph problems such as the independent set problem, network reliability, vertex cover, graph k colorability, and Hamilton circuits. Here and elsewhere [22, 16] we extend the approach to a different set of problems. The following paragraphs ....

S. Arnborg and A. Proskourowski. Linear time algorithms for np-hard problems restricted to partial k-trees. Discrete and Applied Mathematics.

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S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math., 23:11--24, 1989.

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S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23:11-24, 1989.

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S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Applied Mathematics, 23:11--24, 1989.

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S. Arnborg and A. Proskourowski, "Linear time algorithms for NP-hard problems restricted to partial k-trees" Discrete and Applied Mathematics 23 (1989) 11-24.

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ARNBORG, S., AND PROSKUROWSKI, A. Linear time algorithms for np-hard problems restricted to partial k-trees. Discrete Applied Mathematics 23, 1 (1989), 11--24.

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Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11--24, 1989.

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Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math., 23(1):11-24, 1989.

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S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Appl. Math., 23 (1989), pp. 11-24.

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S. Arnborg, A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Appl. Math., 23 (1989), pp. 11-24.

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S. ARNBORG AND A. PROSKUROWSKI, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Appl. Math., 23 (1989), pp. 11--24.

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S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math. 23, 11 - 24, 1989.

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Arnborg S. and Proskurowski A . Linear time algorithms for NP-hard problems restricted to partial k-trees Discrete Applied Math., 23, 1989, 11- 24.

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