Faster Parameterized Algorithms for Minimum Fill-in

We present two parameterized algorithms for the Minimum Fill-in problem, also known as Chordal Completion: given an arbitrary graph G and integer k, can we add at most k edges to G to obtain a chordal graph? Our first algorithm has running time $\mathcal {O}(k^{2}nm+3.0793^{k})$, and requires polynomial space. This improves the base of the exponential part of the best known parameterized algorithm time for this problem so far. We are able to improve this running time even further, at the cost of more space. Our second algorithm has running time $\mathcal {O}(k^{2}nm+2.35965^{k})$ and requires $\mathcal {O}^{\ast}(1.7549^{k})$ space. To achieve these results, we present a new lemma describing the edges that can safely be added to achieve a chordal completion with the minimum number of edges, regardless of k.