In the Interval Completion problem we are given an n-vertex graph G and an integer k, and the task is to transform G by making use of at most k edge additions into an interval graph. This is a fundamental graph modification problem with applications in sparse matrix multiplication and molecular biology. The question about fixed-parameter tractability of Interval Completion was asked by Kaplan, Shamir and Tarjan [FOCS 1994; SIAM J. Comput. 1999] and was answered affirmatively more than a decade later by Villanger at el. [STOC 2007; SIAM J. Comput. 2009], who presented an algorithm with running time O(k 2k n 3 m). We give the first subexponential parameterized algorithm solving Interval Completion in time k O( √ k) n O(1) . This adds Interval Completion to a very small list of parameterized graph modification problems solvable in subexponential time.
In the Proper Interval Completion problem we are given a graph G and an integer k, and the task is to turn G using at most k edge additions into a proper interval graph, i.e., a graph admitting an intersection model of equal-length intervals on a line. The study of Proper Interval Completion from the viewpoint of parameterized complexity has been initiated by Kaplan, Shamir and Tarjan [FOCS 1994; SIAM J. Comput. 1999], who showed an algorithm for the problem working in O(16time. In this paper we present an algorithm with running time k, which is the first subexponential parameterized algorithm for Proper Interval Completion.
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