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Register a new userParameterized complexity in multiple-interval graphs: domination, partition, separation, irredundancy
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Minghui Jiang
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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We show that the problem k-Dominating Set and its several variants including k-Connected Dominating Set, k-Independent Dominating Set, and k-Dominating Clique, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multiple-interval graphs and their complements. This answers an open question of Fellows et al. In sharp contrast, we show that d-Distance k-Dominating Set for d >= 2 is W[2]-complete in multiple-interval graphs and their complements. We also show that k-Perfect Code and d-Distance k-Perfect Code for d >= 2 are W[1]-complete even in unit 2-track interval graphs. In addition, we present various new results on the parameterized complexities of k-Vertex Clique Partition and k-Separating Vertices in multiple-interval graphs and their complements, and present a very simple alternative proof of the W[1]-hardness of k-Irredundant Set in general graphs.
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Butman, Hermelin, Lewenstein, and Rawitz proved that Clique in t-interval graphs is NP-hard for t >= 3. We strengthen this result to show that Clique in 3-track interval graphs is APX-hard.
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A emph{2-interval} is the union of two disjoint intervals on the real line. Two 2-intervals $D_1$ and $D_2$ are emph{disjoint} if their intersection is empty (i.e., no interval of $D_1$ intersects any interval of $D_2$). There can be three different relations between two disjoint 2-intervals; namely, preceding ($<$), nested ($sqsubset$) and crossing ($between$). Two 2-intervals $D_1$ and $D_2$ are called emph{$R$-comparable} for some $Rin{<,sqsubset,between}$, if either $D_1RD_2$ or $D_2RD_1$. A set $mathcal{D}$ of disjoint 2-intervals is $mathcal{R}$-comparable, for some $mathcal{R}subseteq{<,sqsubset,between}$ and $mathcal{R} eqemptyset$, if every pair of 2-intervals in $mathcal{R}$ are $R$-comparable for some $Rinmathcal{R}$. Given a set of 2-intervals and some $mathcal{R}subseteq{<,sqsubset,between}$, the objective of the emph{2-interval pattern problem} is to find a largest subset of 2-intervals that is $mathcal{R}$-comparable. The 2-interval pattern problem is known to be $W[1]$-hard when $|mathcal{R}|=3$ and $NP$-hard when $|mathcal{R}|=2$ (except for $mathcal{R}={<,sqsubset}$, which is solvable in quadratic time). In this paper, we fully settle the parameterized complexity of the problem by showing it to be $W[1]$-hard for both $mathcal{R}={sqsubset,between}$ and $mathcal{R}={<,between}$ (when parameterized by the size of an optimal solution); this answers an open question posed by Vialette [Encyclopedia of Algorithms, 2008].
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