447 research outputs found

    Chordal Editing is Fixed-Parameter Tractable

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    Graph modification problems typically ask for a small set of operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem that allows all three types of operations: given a graph and integers k(1), k(2), and k(3), the CHORDAL EDITING problem asks whether G can be transformed into a chordal graph by at most k(1) vertex deletions, k(2) edge deletions, and k(3) edge additions. Clearly, this problem generalizes both CHORDAL DELETION and CHORDAL COMPLETION (also known as MINIMUM FILL-IN). Our main result is an algorithm for CHORDAL EDITING in time 2(O(klog k)). n(O(1)), where k:=k(1) + k(2) + k(3) and n is the number of vertices of G. Therefore, the problem is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm is both more efficient and conceptually simpler than the previously known algorithm for the special case CHORDAL DELETION

    Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes

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    Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of P4P_4-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parameterized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [21] and computer-aided branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and Applications (DMAA

    Unit Interval Editing is Fixed-Parameter Tractable

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    Given a graph~GG and integers k1k_1, k2k_2, and~k3k_3, the unit interval editing problem asks whether GG can be transformed into a unit interval graph by at most k1k_1 vertex deletions, k2k_2 edge deletions, and k3k_3 edge additions. We give an algorithm solving this problem in time 2O(klog⁡k)⋅(n+m)2^{O(k\log k)}\cdot (n+m), where k:=k1+k2+k3k := k_1 + k_2 + k_3, and n,mn, m denote respectively the numbers of vertices and edges of GG. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time O(4k⋅(n+m))O(4^k \cdot (n + m)). Another result is an O(6k⋅(n+m))O(6^k \cdot (n + m))-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time O(6k⋅n6)O(6^k \cdot n^6).Comment: An extended abstract of this paper has appeared in the proceedings of ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an appendix is provided for a brief overview of related graph classe

    A polynomial kernel for Block Graph Deletion

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    In the Block Graph Deletion problem, we are given a graph GG on nn vertices and a positive integer kk, and the objective is to check whether it is possible to delete at most kk vertices from GG to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with O(k6)\mathcal{O}(k^{6}) vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of `complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time 10k⋅nO(1)10^{k}\cdot n^{\mathcal{O}(1)}.Comment: 22 pages, 2 figures, An extended abstract appeared in IPEC201

    Structural parameterizations for boxicity

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    The boxicity of a graph GG is the least integer dd such that GG has an intersection model of axis-aligned dd-dimensional boxes. Boxicity, the problem of deciding whether a given graph GG has boxicity at most dd, is NP-complete for every fixed d≄2d \ge 2. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive 11-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. that boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page
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