A subexponential parameterized algorithm for Interval Completion

TL;DR

This paper presents the first subexponential fixed-parameter algorithm for the Interval Completion problem, significantly improving the efficiency of transforming a graph into an interval graph with limited edge additions.

Contribution

The authors develop the first subexponential parameterized algorithm for Interval Completion, advancing the computational methods for graph modification problems.

Findings

Achieved a subexponential time algorithm with complexity $k^{O(\sqrt{k})} n^{O(1)}$

Placed Interval Completion among few graph problems solvable in subexponential time

Improved upon previous algorithms with exponential dependence on k

Abstract

In the 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 an interval graph, i.e., a graph admitting an intersection model of intervals on a line. Motivated by applications in sparse matrix multiplication and molecular biology, Kaplan, Shamir and Tarjan [FOCS 1994; SIAM J. Comput. 1999] asked for a fixed-parameter algorithm solving this problem. This question was answer affirmatively more than a decade later by Villanger at el. [STOC 2007; SIAM J. Comput. 2009], who presented an algorithm with running time $O(k^{2k}n^3m)$. We give the first subexponential parameterized algorithm solving Interval Completion in time $k^{O(\sqrt{k})} n^{O(1)}$. This adds Interval Completion to a very small list of parameterized graph modification problems solvable in subexponential time.