A Subexponential Parameterized Algorithm for Proper Interval Completion

Abstract

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 [SIAM J. Comput., 28 (1999), pp. 1906--1922], 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.