Partitive hypergraphs

Abstract. Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS'09) and of Fomin et al. (FOCS'10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use a structural parameters of the input and not the solution size. Let C be a graph class. We define the C-cover number of a graph to be a the smallest number of modules the vertex set can be partitioned into such that each module induces a subgraph that belongs to the class C. We show that each graph problem that can be expressed in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the C-cover number for any fixed class C of bounded rank-width (or equivalently, of bounded clique-width, or bounded Boolean width). Many graph problems such as INDEPENDENT DOMINATING SET, c-COLORING, and c-DOMATIC NUMBER are covered by this meta-kernelization result. Our second result applies to MSO expressible optimization problems, such as MINIMUM VERTEX COVER, MINIMUM DOMINATING SET, and MAXIMUM CLIQUE. We show that these problems admit a polynomial annotated kernel with a linear number of vertices.

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“…The proof of Theorem 1 relies on a combinatorial property of modules of bounded rank-width that amounts to a variant of partitivity [8].…”

Section: Theorem 1 For Every Constant D a Smallest Rank-width-d Covmentioning

confidence: 99%

Meta-kernelization with structural parameters

Ganian

Slivovsky

Szeider

2016

Journal of Computer and System Sciences

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“…Then, it suffices to deal with linear vertices whose parent is linear in the same way than for a definite tree. 3, 2, 1, 13, 15, 14, 16, 8, 9, 10, 11, 12 (1), (13,15,14,16,8,9,10,11,12,5,6,7), (13,15,14,16), (13), (15,14), (14), (16), (8,9,10,11,12), (11), (5, 6, 7).…”

Section: A Computing Perfect Scenarios With Unambiguous Treesmentioning

confidence: 99%

“…Otherwise K \ K (resp. K \ K) would Comparing the rat and mouse X chromosomes with a subset of their common intervals (intervals (13,14,15), (14,15,16), (13,14,15,16), (10,11), (9,10,11), (10,11,12), (8,9,10,11) and (9,10,11,12) were removed from the set of common intervals). The length of a parsimonious perfect scenario with respect to F * .…”

Section: Computing Perfect Scenarios For a Subset Of Common Intervmentioning

confidence: 99%

IEEE/ACM Transactions on Computational Biology and Bioinformatics

2008

IEEE Eng. Med. Biol. Mag.

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This paper proposes new algorithms for computing pairwise rearrangement scenarios that conserve the combinatorial structure of genomes. More precisely, we investigate the problem of sorting signed permutations by reversals without breaking common intervals. We describe a combinatorial framework for this problem that allows to characterize classes of signed permutations for which one can compute in polynomial time a shortest reversal scenario that conserves all common intervals. In particular we define a class of permutations for which this computation can be done in linear time with a very simple algorithm that does not rely on the classical Hannenhalli-Pevzner theory for sorting by reversals. We apply these methods to the computation of rearrangement scenarios between permutations obtained from 16 synteny blocks of the X chromosomes of the human, mouse and rat.

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“…when the permutations are identical. The decomposition addressed in this paper is based on the seminal works on weakly partitive families [9,25]. Let us recall some useful formalisms.…”

Section: Common Interval Decompositionmentioning

confidence: 99%

“…A family F ⊆ 2 V is weakly partitive if and only if ∅ / ∈ F, F contains the trivial subsets (singletons and V ), and F is closed by intersection, union, and differences on overlapping subsets. It is partitive if weakly partitive and closed by symmetric difference on overlapping subsets [9,25]. A weakly partitive family can have O(n!)…”

Section: Combinatorial Decomposition Aspectsmentioning

confidence: 99%

Revisiting T. Uno and M. Yagiura’s Algorithm

Xuan

Habib

Paul

2005

Algorithms and Computation

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Abstract.In 2000, T. Uno and M. Yagiura published an algorithm that computes all the K common intervals of two given permutations of length n in O(n + K) time. Our paper first presents a decomposition approach to obtain a compact encoding for common intervals of d permutations. Then, we revisit T. Uno and M. Yagiura's algorithm to yield a linear time algorithm for finding this encoding. Besides, we adapt the algorithm to obtain a linear time modular decomposition of an undirected graph, and thereby propose a formal invariant-based proof for all these algorithms.

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Partitive hypergraphs

Cited by 70 publications

References 12 publications

Meta-kernelization with structural parameters

Meta-kernelization with structural parameters

IEEE/ACM Transactions on Computational Biology and Bioinformatics

Revisiting T. Uno and M. Yagiura’s Algorithm

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