Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms 2015
DOI: 10.1137/1.9781611974331.ch79
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Lower bounds for the parameterized complexity of Minimum Fill-In and other completion problems

Abstract: In this work, we focus on several completion problems for subclasses of chordal graphs: Minimum Fill-In, Interval Completion, Proper Interval Completion, Threshold Completion, and Trivially Perfect Completion. In these problems, the task is to add at most k edges to a given graph in order to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related Chain Completion problem:• Assuming t… Show more

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Cited by 10 publications
(14 citation statements)
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“…Similar idea had actually been used by Kashiwabara and Fujisawa [22,23]. All hitherto known reductions on related problems, including [5,6,36], followed this idea, and used graph layout problems as the source problems. An important benefit of this approach is that they can be (usually in an effortless way) applied to related problems on (proper) interval graphs.…”
Section: Discussionmentioning
confidence: 99%
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“…Similar idea had actually been used by Kashiwabara and Fujisawa [22,23]. All hitherto known reductions on related problems, including [5,6,36], followed this idea, and used graph layout problems as the source problems. An important benefit of this approach is that they can be (usually in an effortless way) applied to related problems on (proper) interval graphs.…”
Section: Discussionmentioning
confidence: 99%
“…Yannakakis' reduction is far more powerful in this sense. It directly applies to interval graphs, unit interval graphs, and strongly chordal graph, while a slight modification works for trivially perfect graphs and threshold graphs (see, e.g., Bliznets et al [6] for details). Interval graphs are the intersection graphs of intervals on the real line.…”
Section: Discussionmentioning
confidence: 99%
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“…Another interesting direction is to investigate further the complexity of graph modification problems related to cutwidth: apply at most k modifications to the given semi-complete digraph in order to obtain a digraph of cutwidth at most c. Some immediate corollaries for vertex deletions are discussed in Section 5, but it is also interesting to look at the arc reversal variant, where the allowed modification is reversing an arc. For c = 0, this problem is equivalent to the Feedback Arc Set problem, which has been studied intensively in tournaments and semi-complete digraphs [1,4,10,11]. Further results on the vertex deletion and arc reversal problems related to cutwidth in semi-complete digraphs will be the topic of a future paper, currently under preparation.…”
Section: Discussionmentioning
confidence: 99%