Subexponential Parameterized Algorithm for I nterval C ompletion

Abstract

In the I nterval C ompletion 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 I nterval C ompletion was asked by Kaplan et al. [FOCS 1994; SIAM J. Comput. 1999] and was answered affirmatively more than a decade later by Villanger et al. [STOC 2007; SIAM J. Comput. 2009], who presented an algorithm with running time O ( k 2 k n 3 m ). We give the first subexponential parameterized algorithm solving I nterval C ompletion in time k O (√ k ) n O (1) . This adds I nterval C ompletion to a very small list of parameterized graph modification problems solvable in subexponential time.