Unit Interval Editing is Fixed-Parameter Tractable

Given a graph G and integers k1, k2, and k3, the unit interval editing problem asks whether G can be transformed into a unit interval graph by at most k1 vertex deletions, k2 edge deletions, and k3 edge additions. We give an algorithm solving this problem in time 2 logk) · (n+m), where k := k1 +k2 +k3, and n,m denote respectively the numbers of vertices and edges of G. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time O(4 · (n+m)). Another result is an O(6 · (n+m))-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van ’t Hof and Villanger, which runs in time O(6 · n6).