Abstract
We study algorithmic properties of the graph class {textsc {Chordal}}{-ke}, 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}. More precisely, we identify a large class of optimization problems on {textsc {Chordal}}{-ke} solvable in time 2^{{mathcal{O}}(sqrt{k}log k)}cdot n^{{mathcal{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} 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} 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} 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} graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of {textsc {Chordal}}{-ke}, namely, {textsc {Interval}}{-ke} and {textsc {Split}}{-ke} graphs.
Highlights
Many -hard graph optimization problems are solvable in polynomial or even linear time when the problem’s input is restricted to a special graph class
We show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on CHORDAL−ke when parameterized by k but do not admit subexponential in k algorithms unless Exponential Time Hypothesis (ETH) fails
For unweighted Independent Set we show that the problem admits polynomial in k kernel on graph classes INTERVAL−ke and SPLIT−ke
Summary
Many -hard graph optimization problems are solvable in polynomial or even linear time when the problem’s input is restricted to a special graph class. We obtain several algorithmic results on the parameterized complexity of optimization problems on F − ke , where F is the class of chordal graphs It follows almost directly from the previous work of Bodlaender, Jansen, and Kratsch [8, 42] that Weighted Independent Set, Weighted Vertex Cover, Weighted Bipartite Subgraph, Weighted Odd Cycle Transversal, Weighted Feedback Vertex Set and Weighted Clique do not admit a polynomial in k kernel (unless coNP ⊈ NP∕poly ) on COMPLETE−ke and on CHORDAL−ke These lower bounds do not refute the possibility of polynomial Turing kernelization or kernelization for unweighted variants of the problems. Jansen and Kratsch [41] proved that for every fixed integer d, the problems d -Coloring and d -List Coloring admit polynomial kernels on the parameterized graph classes SPLIT + kv , COCHORDAL + kv , and COGRAPH + kv.
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