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 the problem in 2O(klogk) ·(n+m) time, 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. This 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(4k · (n + m)). Another result is an O(6k · (n + m))-time algorithm for the unit interval vertex deletion problem, significantly improving the best-known algorithm running in time O(6k · n6).