Exploring Subexponential Parameterized Complexity of Completion Problems (1309.4022v2)

Abstract: Let ${\cal F}$ be a family of graphs. In the ${\cal F}$-Completion problem, we are given a 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 ${\cal F}$ as an induced subgraph. It appeared recently that special cases of ${\cal F}$-Completion, the problem of completing into a chordal graph known as Minimum Fill-in, corresponding to the case of ${\cal F}={C_4,C_5,C_6,\ldots}$, and the problem of completing into a split graph, i.e., the case of ${\cal F}={C_4, 2K_2, C_5}$, are solvable in parameterized subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$. The exploration of this phenomenon is the main motivation for our research on ${\cal F}$-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 $2^{O(\sqrt{k}\log{k})}n^{O(1)}$, that is ${\cal F}$-Completion for ${\cal F} ={C_4, P_4}$, a cycle and a path on four vertices. - The problems known in the literature as Pseudosplit Completion, the case where ${\cal F} = {2K_2, C_4}$, and Threshold Completion, where ${\cal F} = {2K_2, P_4, C_4}$, are also solvable in time $2^{O(\sqrt{k}\log{k})} n^{O(1)}$. We complement our algorithms for ${\cal F}$-Completion with the following lower bounds: - For ${\cal F} = {2K_2}$, ${\cal F} = {C_4}$, ${\cal F} = {P_4}$, and ${\cal F} = {2K_2, P_4}$, ${\cal F}$-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 ${\cal F}$-Completion problems for ${\cal F}\subseteq{2K_2, C_4, P_4}$.