Modular-Width: An Auxiliary Parameter for Parameterized Parallel Complexity

Abu-Khzam, Faisal N.; Li, Shouwei; Markarian, Christine; Heide, Friedhelm Meyer auf der; Podlipyan, Pavel

doi:10.1007/978-3-319-59605-1_13

Cited by 7 publications

(38 citation statements)

References 18 publications

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“…We start from two recently introduced parameters: modular-width [22] and neighborhood diversity [31]. Both parameters received much attention [1,2,5,7,12,17,18,21,24,25,29] also due to their property of being computable in polynomial time [22,31].…”

Section: Introductionmentioning

confidence: 99%

Iterated Type Partitions

Cordasco

Gargano

Rescigno

2020

Lecture Notes in Computer Science

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This paper introduces a novel parameter, called iterated type partition, that can be computed in polynomial time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parametrized by the iterated type partition. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parametrized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity. Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition, which enables us to find optimal solutions for several graph problems. While the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. As an example, in this paper, we give an algorithm for the Dominating Set problem that outputs an optimal set in time O(2 t + poly(n)), where n and t are the size and the iterated type partition of the input graph, respectively.

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Section: Introductionmentioning

confidence: 99%

Iterated Type Partitions

Cordasco

Gargano

Rescigno

2020

Lecture Notes in Computer Science

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“…2 2 Λ ). We note in passing that Λ is always less than or equal to the maximum degree in the modular decomposition tree, which is also known as modular-width [1,18]. Hence, the latter findings together with Lemma 5.7 imply the following Observation 5.8.…”

Section: A Modular-decomposition-based Heuristic For Cograph Editingmentioning

confidence: 65%

“…Therefore, assume that M P M i+1 . The module M cannot overlap M i+1 , since M i+1 is strong in H. As shown above, the case M i+1 M P M i+1 cannot occur, and thus we have either (1)…”

Section: Pairwise Module Merge and Algorithmic Issuesmentioning

confidence: 86%

“…We provide an exact algorithm that allows to optimally edit a cograph via pairwise module-merges. As by-product, we obtain an FPT algorithm for the case that cograph editing is parameterized by the so-called modular-width [1,18]. We finish this paper with a short discussion on how the latter method can be used to obtain a simple O(|V | 2 )-time heuristic.…”

Section: Introductionmentioning

confidence: 99%

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Cograph editing: Merging modules is equivalent to editing P_4s

Fritz

Hellmuth

Stadler

et al. 2020

Art Discrete Appl. Math.

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The modular decomposition of a graph G = (V, E) does not contain prime modules if and only if G is a cograph, that is, if no quadruple of vertices induces a simple connected path P 4 . The cograph editing problem consists in inserting into and deleting from G a set F of edges so that H = (V, E F ) is a cograph and |F | is minimum. This NP-hard combinatorial optimization problem has recently found applications, e.g., in the context of phylogenetics. Efficient heuristics are hence of practical importance. The simple characterization of cographs in terms of their modular decomposition suggests that instead of editing G one could operate directly on the modular decomposition. We show here that editing the * We thank the anonymous referees for their helpful comments. This work was funded in part by the Deutsche Forschungsgemeinschaft (DFG), Proj. Nr. 432974470 (to PFS). † PFS is also affiliated with the c b This work is licensed under https://creativecommons.org/licenses/by/4.0/ Art Discrete Appl. Math. 3 (2020) #P2.01induced P 4 s is equivalent to resolving prime modules by means of a suitable defined merge operation on the submodules. Moreover, we characterize so-called module-preserving edit sets and demonstrate that optimal pairwise sequences of module-preserving edit sets exist for every non-cograph. This eventually leads to an exact algorithm for the cograph editing problem as well as fixed-parameter tractable (FPT) results when cograph editing is parameterized by the so-called modular-width. In addition, we provide two heuristics with time complexity O(|V | 3 ), resp., O(|V | 2 ).

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“…More recently, the work reported in [18] adopted the graph modular width as auxiliary parameter. The Weighted Maximum Clique and the Maximum Matching problems were shown to be solvable in O(2 k * log(n)) using O(n) processors, when the modular-width is bounded by k.…”

Section: Key Results and Methodsmentioning

confidence: 99%

A Brief Survey of Fixed-Parameter Parallelism

Abu-Khzam

Kontar

2020

Algorithms

Self Cite

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This paper provides an overview of the field of parameterized parallel complexity by surveying previous work in addition to presenting a few new observations and exploring potential new directions. In particular, we present a general view of how known FPT techniques, such as bounded search trees, color coding, kernelization, and iterative compression, can be modified to produce fixed-parameter parallel algorithms.

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Modular-Width: An Auxiliary Parameter for Parameterized Parallel Complexity

Cited by 7 publications

References 18 publications

Iterated Type Partitions

Iterated Type Partitions

Cograph editing: Merging modules is equivalent to editing P_4s

A Brief Survey of Fixed-Parameter Parallelism

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