Pål Grønås Drange
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 was shown recently that two special cases of F-Completion, namely, (i) the problem of completing into a chordal graph known as Minimum Fill-in (SIAM J. Comput. 2013), which corresponds to the case of F={C4, C5, C6, …}, and (ii) the problem of completing into a split graph (Algorithmica 2015), that is, the case of F={C4, 2K2, C5}, are solvable in parameterized subexponential time 2O(√klogk)nO(1). The exploration of this phenomenon is the main motivation for our research on F-Completion.
In this article, we prove that completions into several well-studied classes of graphs without long induced cycles and paths also admit parameterized subexponential time algorithms by showing that:
—The problem Trivially Perfect Completion, which is F-Completion for F={C4, P4}, a cycle and a path on four vertices, is solvable in parameterized subexponential time 2O(√klogk)nO(1).
—The problems known in the literature as Pseudosplit Completion, the case in which F{2K2, C4}, and Threshold Completion, in which F=2K2, P4, C4}, are also solvable in time 2O(√klogk)nO}(1).
We complement our algorithms for F-Completion with the following lower bounds:
—For F={2K2}, F= {C4}, F={Po4}, and F={2K2, P4}, F-Completion cannot be solved in time 2o(k)nO(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 any F ⊆ {2K2, C4, P4}.
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Index Terms
- Exploring the Subexponential Complexity of Completion Problems
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