Lower bounds for the parameterized complexity of Minimum Fill-In and other completion problems
In this work, we focus on several completion problems for subclasses of chordal graphs: Minimum Fill-In, Interval Completion, Proper IntervalCompletion, Threshold Completion, and Trivially Perfect Completion. In these problems, the task is to add at most k edges to a given graph in order to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related Chain Completion problem:
• Assuming the Exponential Time Hypothesis, none of these problems can be solved in time [EQUATION] or [EQUATION], (for some integer c.
• Assuming the non-existence of a subexponential-time approximation scheme for Min Bisection on d-regular graphs, for some constant d, none of these problems can be solved in time 2o(n) or 2o([EQUATION]) · nO(1).
The second result is an evidence, that a significant improvement of the best known algorithms for parameterized completion problems would lead to a surprising breakthrough in the design of approximation algorithms for Min Bisection.