996 research outputs found

    Parameterized Algorithms for Modular-Width

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    It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty. The main contribution of this paper is to consider a parameter called modular-width, defined using the well-known notion of modular decompositions. Using a combination of ILPs and dynamic programming we manage to design FPT algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path and Hamiltonian cycle), which are W-hard for both clique-width and its recently introduced restriction, shrub-depth. We thus argue that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with arXiv:1304.5479 by other author

    Exploiting c\mathbf{c}-Closure in Kernelization Algorithms for Graph Problems

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    A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show

    Improved FPT algorithms for weighted independent set in bull-free graphs

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    Very recently, Thomass\'e, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time 2O(k5)n92^{O(k^5)} \cdot n^9. In this article we improve this running time to 2O(k2)n72^{O(k^2)} \cdot n^7. As a byproduct, we also improve the previous Turing-kernel for this problem from O(k5)O(k^5) to O(k2)O(k^2). Furthermore, for the subclass of bull-free graphs without holes of length at most 2p12p-1 for p3p \geq 3, we speed up the running time to 2O(kk1p1)n72^{O(k \cdot k^{\frac{1}{p-1}})} \cdot n^7. As pp grows, this running time is asymptotically tight in terms of kk, since we prove that for each integer p3p \geq 3, Weighted Independent Set cannot be solved in time 2o(k)nO(1)2^{o(k)} \cdot n^{O(1)} in the class of {bull,C4,,C2p1}\{bull,C_4,\ldots,C_{2p-1}\}-free graphs unless the ETH fails.Comment: 15 page

    Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

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    Given a graph GG and a parameter kk, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset UV(G)U\subseteq V(G) of size at most kk that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size O(k161log58k)O(k^{161}\log^{58}k), and asked whether one can design a kernel of size O(k10)O(k^{10}). While we do not completely resolve this question, we design a significantly smaller kernel of size O(k12log10k)O(k^{12}\log^{10}k), inspired by the O(k2)O(k^2)-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution
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