Parameterized Algorithms for Modular-Width

Gajarský, Jakub; Lampis, Michael; Ordyniak, Sebastian

doi:10.1007/978-3-319-03898-8_15

Cited by 75 publications

(70 citation statements)

References 27 publications

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“…The class of bounded tree-length graphs is an extremely rich graph class as it contains several well-studied graph classes like interval graphs, chordal graphs, AT-free graphs, permutation graphs and so on. Modular-width is a larger parameter than the more general clique-width and has been used in the past as a parameterization for problems where choosing clique-width as a parameter leads to W-hardness [10]. This provides a strong motivation for studying the role played by the tree-length of a graph in the computation of its metric dimension.…”

Section: Metric Dimensionmentioning

confidence: 99%

Metric Dimension of Bounded Width Graphs

Belmonte

Fomin

Golovach

et al. 2015

Mathematical Foundations of Computer Science 2015

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The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y. A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph. The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes. In recent work, Foucaud et al. (2015) showed that this problem is NP-complete even on interval graphs. They complemented this result by also showing that it is fixed-parameter tractable (FPT) parameterized by the metric dimension of the graph. In this work, we show that this FPT result can in fact be extended to all graphs of bounded tree-length. This includes well-known classes like chordal graphs, AT-free graphs and permutation graphs. We also show that this problem is FPT parameterized by the modular-width of the input graph.

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Section: Metric Dimensionmentioning

confidence: 99%

Metric Dimension of Bounded Width Graphs

Belmonte

Fomin

Golovach

et al. 2015

Mathematical Foundations of Computer Science 2015

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show abstract

“…This problem was shown to be W [1]-hard by Fellows et al [9] when parameterized by the tree-width of the input graph. A strengthening of this fact-the Equitable Coloring problem is W [1]-hard with respect to tree-depth was proven by Gajarský et al [11]. We would like to ask a question concerning these graph models, namely the tree-depth.…”

Section: Conjecture 62 For Every Disconnected Graph H the Partitionmentioning

confidence: 97%

“…Both previous approaches are generalized by a modular-width, defined by Gajarský et al [11]. Here we deal with graphs created by an algebraic expression that uses the following operations: 1. create an isolated vertex, 2. the disjoint union of two graphs, that is from graphs…”

Section: Definition 32 (Neighborhood Diversity [19])mentioning

confidence: 99%

Partitioning Graphs into Induced Subgraphs

Knop

2017

Language and Automata Theory and Applications

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We study the Partition into H problem from the parameterized complexity point of view. In the Partition into H problem the task is to partition vertices of a graph G into sets V 1 , V 2 , . . . , V n such that the graph H is isomorphic to the subgraph of G induced by each set V i for i = 1, 2, . . . , n. The pattern graph H is fixed.For the parametrization we consider three distinct structural parameters of the graph Gnamely the tree-width, the neighborhood diversity, and the modular-width. For the parametrization by the neighborhood diversity we obtain an FPT algorithm for every graph H. For the parametrization by the tree-width we obtain an FPT algorithm for every connected graph H. Finally, for the parametrization by the modular-width we derive an FPT algorithm for every prime graph H.

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“…In the recent paper [7] Gajarský et al give an FPT algorithm (but not EPT) for Hamiltonian Cycle parameterized by modular-width, and show W-hardness when parameterized by shrub-depth. Split-matching-width is a new parameter introduced by Saether and Telle [13] for which Hamiltonian Cycle is FPT [13].…”

Section: Introductionmentioning

confidence: 99%

“…A directed path from parameter a to b means that there exists a function f so that for all graphs G, b(G) ≤ f (a(G)). The arrows related to sm-width are from [13], for the others see Gajarský et al [7]. The colors depict the complexity of Hamiltonian Cycle parameterized with the respective parameters.…”

Section: Introductionmentioning

confidence: 99%

Solving Hamiltonian Cycle by an EPT algorithm for a non-sparse parameter

Sæther

2017

Discrete Applied Mathematics

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Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter cliquewidth, Hamiltonian Cycle becomes W[1]-hard, as shown by Fomin et al. [5]. Saether and Telle address this problem in their paper [13] by introducing a new parameter, split-matching-width, which lies between treewidth and clique-width in terms of generality. They show that even though graphs of restricted split-matching-width might be dense, solving problems such as Hamiltonian Cycle can be done in FPT time.Recently, it was shown that Hamiltonian Cycle parameterized by treewidth is in EPT [1,6], meaning it can be solved in n O(1) 2 O(k) -time. In this paper, using tools from [6], we show that also parameterized by split-matching-width Hamiltonian Cycle is EPT. To the best of our knowledge, this is the first EPT algorithm for any "globally constrained" graph problem parameterized by a non-trivial and non-sparse structural parameter. To accomplish this, we also give an algorithm constructing a branch decomposition approximating the minimum split-matching-width to within a constant factor. Combined, these results show that the algorithms in [13] for Edge Dominating Set, Chromatic Number and Max Cut all can be improved. We also show that for Hamiltonian Cycle and Max Cut the resulting algorithms are asymptotically optimal under the Exponential Time Hypothesis.

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Parameterized Algorithms for Modular-Width

Cited by 75 publications

References 27 publications

Metric Dimension of Bounded Width Graphs

Metric Dimension of Bounded Width Graphs

Partitioning Graphs into Induced Subgraphs

Solving Hamiltonian Cycle by an EPT algorithm for a non-sparse parameter

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