Chordal Editing is Fixed-Parameter Tractable

Graph modification problems are typically asked as follows: is there a small set of operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem which allows all three types of operations: given a graph $G$ and integers $k1$, $k2$, and $k3$, the \textsc{chordal editing} problem asks whether $G$ can be transformed into a chordal graph by at most $k1$ vertex deletions, $k2$ edge deletions, and $k3$ edge additions. Clearly, this problem generalizes both \textsc{chordal vertex/edge deletion} and \textsc{chordal completion} (also known as \textsc{minimum fill-in}). Our main result is an algorithm for \textsc{chordal editing} in time $2^{O(k\log k)}\cdot n^{O(1)}$, where $k:=k1+k2+k_3$ and $n$ is the number of vertices of $G$. Therefore, the problem is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm is both more efficient and conceptually simpler than the previously known algorithm for the special case \textsc{chordal deletion}.