A Subexponential Parameterized Algorithm for Proper Interval Completion

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 \(\mathcal{O}(16^k\cdot (n+m))\) time. In this paper we present an algorithm with running time \(k^{\mathcal{O}(k^{2/3})} + \mathcal{O}(nm(kn+m))\), which is the first subexponential parameterized algorithm for Proper Interval Completion.

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