44 research outputs found

    Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

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    The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasability of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Date) architectures.Comment: Added experiments on one more data se

    Parameterized analysis of complexity

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    On the Descriptive Complexity of Color Coding

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    Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing - purely in terms of the syntactic structure of describing formulas - when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in fpt just because of the way they are commonly described using logical formulas

    Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph

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    We present fixed-parameter tractable (FPT) algorithms for two problems, Maximum Happy Set (MaxHS) and Maximum Edge Happy Set (MaxEHS)--also known as Densest k-Subgraph. Given a graph GG and an integer kk, MaxHS asks for a set SS of kk vertices such that the number of happy vertices\textit{happy vertices} with respect to SS is maximized, where a vertex vv is happy if vv and all its neighbors are in SS. We show that MaxHS can be solved in time O(2mw⋅mw⋅k2⋅∣V(G)∣)\mathcal{O}\left(2^\textsf{mw} \cdot \textsf{mw} \cdot k^2 \cdot |V(G)|\right) and O(8cw⋅k2⋅∣V(G)∣)\mathcal{O}\left(8^\textsf{cw} \cdot k^2 \cdot |V(G)|\right), where mw\textsf{mw} and cw\textsf{cw} denote the modular-width\textit{modular-width} and the clique-width\textit{clique-width} of GG, respectively. This resolves the open questions posed in literature. The MaxEHS problem is an edge-variant of MaxHS, where we maximize the number of happy edges\textit{happy edges}, the edges whose endpoints are in SS. In this paper we show that MaxEHS can be solved in time f(nd)⋅∣V(G)∣O(1)f(\textsf{nd})\cdot|V(G)|^{\mathcal{O}(1)} and O(2cd⋅k2⋅∣V(G)∣)\mathcal{O}\left(2^{\textsf{cd}}\cdot k^2 \cdot |V(G)|\right), where nd\textsf{nd} and cd\textsf{cd} denote the neighborhood diversity\textit{neighborhood diversity} and the cluster deletion number\textit{cluster deletion number} of GG, respectively, and ff is some computable function. This result implies that MaxEHS is also fixed-parameter tractable by twin cover number\textit{twin cover number}
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