Interval Completion Is Fixed Parameter Tractable

We present an algorithm with runtime O(knm) for the following NP-complete problem [9, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the long-standing open question [17, 7, 25, 14], first posed by Kaplan, Shamir and Tarjan, of whether this problem was fixed parameter tractable. The problem has applications in Profile Minimization for Sparse Matrix Computations [10, 26], and our results show tractability for the case of a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search.