Deciding Clique-Width for Graphs of Bounded Tree-Width

Espelage, Wolfgang; Gurski, Frank; Wanke, Egon

doi:10.1007/3-540-44634-6_9

Cited by 21 publications

(20 citation statements)

References 12 publications

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“…Consequently, we use μ(T 1 , T 2 ) to denote a joint operator that applies a sequence of ⊕, η i,j , or ρ i → j operations to serve as a merging operator. This joint operator is motivated by the canonical form of the clique-width expression [26], in which after each disjoint union ⊕, first, several edge introductions η i,j and then several relabeling operations ρ i→j must occur [20].…”

Section: The K-ambiguity Treementioning

confidence: 99%

Backtracking-based dynamic programming for resolving transmit ambiguities in WSN localization

Schlupkothen

Prasse

Ascheid

2018

EURASIP J. Adv. Signal Process.

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The complexity of agent localization increases significantly when unique identification of the agents is not possible. Corresponding application cases include multiple-source localization, in which the agents do not have identification sequences at all, and scenarios in which it is infeasible to send sufficiently long identification sequences, e.g., for highly resource-limited agents. The complexity increase is due to the need to solve an additional optimization problem to resolve the indifferentiability of the agents and thus to enable their localization. In this work, we present a thorough analysis of this problem and propose a maximum a posteriori (MAP)-optimal algorithm based on graph decompositions and expression trees. The proposed algorithm efficiently exploits the fixed-parameter tractability of the underlying graph-theoretical problem and employs dynamic programming and backtracking. We show that the proposed algorithm is able to reduce the run time by up to 88.3% compared with a corresponding MAP-optimal integer linear programming formulation.

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Section: The K-ambiguity Treementioning

confidence: 99%

Backtracking-based dynamic programming for resolving transmit ambiguities in WSN localization

Schlupkothen

Prasse

Ascheid

2018

EURASIP J. Adv. Signal Process.

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“…However it can be shown since the upper bound proof in [5] is constructive (see also [8,14]). Note that if a graph has bounded tree-width then the corresponding tree decomposition can be constructed in linear time [2].…”

Section: Theorem 21 ([5])mentioning

confidence: 99%

Intractability of Clique-Width Parameterizations

Fomin

Golovach²,

Lokshtanov³

et al. 2010

SIAM J. Comput.

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Abstract. We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]-hard parameterized by clique-width. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the clique-width, that is, solvable in time g(k) · n O(1) on n-vertex graphs of clique-width k, where g is some function of k only. Our results imply that the running time O(n f (k) ) of many clique-width based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namelyKey words. Parameterized complexity, clique-width, tree-width, chromatic number, edge domination, hamiltonian cycle AMS subject classifications. 68Q17, 68Q25, 68W401. Introduction. One of the most frequent approaches for solving graph problems is based on decomposition methods. Tree decomposition, and the corresponding parameter, the tree-width of a graph, are among the most commonly used concepts. We refer to the surveys of Bodlaender [3] and Hlinený et al. [22] for further references on tree-width and related parameters. In the quest for alternative graph decompositions that can be applied to broader classes than to those of bounded tree-width and still enjoy good algorithmic properties, Courcelle and Olariu [10] introduced the clique-width of a graph. Clique-width can be seen as a generalization of tree-width, in a sense that every graph class of bounded tree-width also have bounded clique-width [5].In recent years, clique-width has received much attention. Corneil, Habib, Lanlignel, Reed, and Rotics [4] show that graphs of clique-width at most 3 can be recognized in polynomial time. 3 ) and computing (2 k+1 − 1)-expressions for a graph G of clique-width at most k. It is also worth to mention here the related graph parameters NLC-width introduced by Wanke [30] and rank-width introduced by Oum and Seymour [27], which are equivalent to clique-width in the sense that the same classes of graph have bounded clique-width, NLC-width ans rank-width.By the seminal result of Courcelle [6, 9] (see also [1]), every decision problem on graphs expressible in monadic second order logic is fixed parameter tractable when parameterized by the tree-width of the input graph. For problems expressible in monadic second order logic with logical formulas that do not use edge set quantifications (so-called M S 1 -logic), it is possible to extend the meta theorem of Courcelle to graphs of bounded clique-width. As it was shown by Courcelle, Makowsky, and

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“…However it can be shown since the upper bound proof in [5] is constructive (see also [8,13]). Note that if a graph has bounded treewidth then the corresponding tree decomposition can be constructed in linear time [2].…”

Section: Theorem 1 ([5])mentioning

confidence: 99%

Clique-width: On the Price of Generality

Fomin¹,

Golovach²,

Lokshtanov³

et al. 2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Many hard problems can be solved efficiently when the input is restricted to graphs of bounded treewidth. By the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the treewidth of the input graph. Moreover, for every fixed k ≥ 0, such problems can be solved in linear time on graphs of treewidth at most k. In particular, this implies that basic problems like Dominating Set, Graph Coloring, Clique, and Hamiltonian Cycle are solvable in linear time on graphs of bounded treewidth.A significant amount of research in graph algorithms has been devoted to extending this result to larger classes of graphs. It was shown that some of the algorithmic meta-theorems for treewidth can be carried over to graphs of bounded clique-width. Courcelle, Makowsky, and Rotics proved that the analogue of Courcelle's result holds for graphs of bounded clique-width when the logical formulas do not use edge set quantifications. Despite of its generality, this does not resolve the parameterized complexity of many basic problems concerning edge subsets (like Edge Dominating Set), vertex partitioning (like Graph Coloring), or global connectivity (like Hamiltonian Cycle). There are various algorithms solving some of these problems in polynomial time on graphs of clique-width at most k. However, these are not fixed parameter tractable algorithms and have typical running times O(n f (k) ), where n is the input length and f is some function.It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the clique-width, i.e. solvable in time O(g(k) · n c ), for some function g and a constant c not depending on k. In this paper we resolve this problem by showing that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]-hard parameterized by clique-width. This shows that the running time O(n f (k) ) of many clique-width based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namely F P T = W [1])-the price we pay for generality.

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Deciding Clique-Width for Graphs of Bounded Tree-Width

Cited by 21 publications

References 12 publications

Backtracking-based dynamic programming for resolving transmit ambiguities in WSN localization

Backtracking-based dynamic programming for resolving transmit ambiguities in WSN localization

Intractability of Clique-Width Parameterizations

Clique-width: On the Price of Generality

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