58 research outputs found

    Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems

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    In the {\sc Hitting Set} problem, we are given a collection F\cal F of subsets of a ground set VV and an integer pp, and asked whether VV has a pp-element subset that intersects each set in F\cal F. We consider two parameterizations of {\sc Hitting Set} below tight upper bounds: p=m−kp=m-k and p=n−kp=n-k. In both cases kk is the parameter. We prove that the first parameterization is fixed-parameter tractable, but has no polynomial kernel unless coNP⊆\subseteqNP/poly. The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph H=(V,F)H=(V,{\cal F}), makes the problem not only fixed-parameter tractable, but also one with a linear kernel. Here the degeneracy of H=(V,F)H=(V,{\cal F}) is the minimum integer dd such that for each X⊂VX\subset V the hypergraph with vertex set V∖XV\setminus X and edge set containing all edges of F\cal F without vertices in XX, has a vertex of degree at most d.d. In {\sc Nonblocker} ({\sc Directed Nonblocker}), we are given an undirected graph (a directed graph) GG on nn vertices and an integer kk, and asked whether GG has a set XX of n−kn-k vertices such that for each vertex y∉Xy\not\in X there is an edge (arc) from a vertex in XX to yy. {\sc Nonblocker} can be viewed as a special case of {\sc Directed Nonblocker} (replace an undirected graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that {\sc Nonblocker} has a linear-order kernel. We obtain a linear-order kernel for {\sc Directed Nonblocker}

    Classes of Intersection Digraphs with Good Algorithmic Properties

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    While intersection graphs play a central role in the algorithmic analysis of hard problems on undirected graphs, the role of intersection digraphs in algorithms is much less understood. We present several contributions towards a better understanding of the algorithmic treatment of intersection digraphs. First, we introduce natural classes of intersection digraphs that generalize several classes studied in the literature. Second, we define the directed locally checkable vertex (DLCV) problems, which capture many well-studied problems on digraphs such as (Independent) Dominating Set, Kernel, and H-Homomorphism. Third, we give a new width measure of digraphs, bi-mim-width, and show that the DLCV problems are polynomial-time solvable when we are provided a decomposition of small bi-mim-width. Fourth, we show that several classes of intersection digraphs have bounded bi-mim-width, implying that we can solve all DLCV problems on these classes in polynomial time given an intersection representation of the input digraph. We identify reflexivity as a useful condition to obtain intersection digraph classes of bounded bi-mim-width, and therefore to obtain positive algorithmic results

    Proceedings of the 17th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

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