Subexponential Parameterized Algorithm for Minimum Fill-In

Abstract

The Minimum Fill-in problem is used to decide if a graph can be triangulated by adding at most $k$ edges. In 1994, Kaplan, Shamir, and Tarjan showed that the problem is solvable in time $\mathcal{O}(2^{\mathcal{O}({k})}+k^2 nm)$ on graphs with $n$ vertices and $m$ edges and thus is fixed parameter tractable. Here, we give the first subexponential parameterized algorithm solving Minimum Fill-in in time $\mathcal{O}(2^{\mathcal{O}(\sqrt{k}\log{k})} +k^2 nm)$. This substantially lowers the complexity of the problem. Techniques developed for Minimum Fill-in can be used to obtain subexponential parameterized algorithms for several related problems, including Minimum Chain Completion, Chordal Graph Sandwich, and Triangulating Colored Graph.