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We study algorithmic properties of the graph class $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e . More precisely, we identify a large class of optimization problems on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e solvable in time $$2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}$$ 2 O ( k log k ) · n O ( 1 ) . Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , namely, $${\textsc {Interval}}{-ke}$$ I N T E R V A L - k e and $${\textsc {Split}}{-ke}$$ S P L I T - k e graphs.
We study algorithmic properties of the graph class $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e . More precisely, we identify a large class of optimization problems on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e solvable in time $$2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}$$ 2 O ( k log k ) · n O ( 1 ) . Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , namely, $${\textsc {Interval}}{-ke}$$ I N T E R V A L - k e and $${\textsc {Split}}{-ke}$$ S P L I T - k e graphs.
We study the problem of deleting the smallest set S of vertices (resp. edges) from a given graph G such that the induced subgraph (resp. subgraph) G \ S belongs to some class H. We consider the case where graphs in H have treewidth bounded by t, and give a general framework to obtain approximation algorithms for both vertex and edge-deletion settings from approximation algorithms for certain natural graph partitioning problems called k-Subset Vertex Separator and k-Subset Edge Separator, respectively.For the vertex deletion setting, our framework combined with the current best result for k-Subset Vertex Separator, improves approximation ratios for basic problems such as k-Treewidth Vertex Deletion and Planar-F Vertex Deletion. Our algorithms are simpler than previous works and give the first deterministic and uniform approximation algorithms under the natural parameterization.For the edge deletion setting, we give improved approximation algorithms for k-Subset Edge Separator combining ideas from LP relaxations and important separators. We present their applications in bounded-degree graphs, and also give an APX-hardness result for the edge deletion problems.3 The conference version of [Lee18] presents a randomized algorithm, but the journal version derandomized it.
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