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Exploring Subexponential Parameterized Complexity of Completion Problems

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Abstract

Let F be a family of graphs. In the F-Completion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k edges can be added to G so that the resulting graph does not contain a graph from F as an induced subgraph. It appeared recently that special cases of F-Completion, the problem of completing into a chordal graph known as "Minimum Fill-in", corresponding to the case of F={C_4,C_5,C_6,...}, and the problem of completing into a split graph, i.e., the case of F={C_4,2K_2,C_5}, are solvable in parameterized subexponential time. The exploration of this phenomenon is the main motivation for our research on F-Completion. In this paper we prove that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms by showing that: - The problem Trivially Perfect Completion is solvable in parameterized subexponential time, that is F-Completion for F={C_4,P_4}, a cycle and a path on four vertices. - The problems known in the literature as Pseudosplit Completion, the case where F={2K_2,C_4}, and Threshold Completion, where F={2K_2,P_4,C_4}, are also solvable in subexponential time. We complement our algorithms for \$F\$-Completion with the following lower bounds: - For F={2K_2}, F={C_4}, F={P_4}, and F={2K_2,P_4}, F-Completion cannot be solved in time 2^o(k).n^O(1) unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of F-Completion problems for F contained inside {2K_2,C_4,P_4}.

BibTeX - Entry

```@InProceedings{drange_et_al:LIPIcs:2014:4465,
author =	{Pal Gronas Drange and Fedor V. Fomin and Michal Pilipczuk and Yngve Villanger},
title =	{{Exploring Subexponential Parameterized Complexity of Completion Problems}},
booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
pages =	{288--299},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-65-1},
ISSN =	{1868-8969},
year =	{2014},
volume =	{25},
editor =	{Ernst W. Mayr and Natacha Portier},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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