Lower bounds for the parameterized complexity of Minimum Fill-In and other completion problems

Abstract

In this work, we focus on several completion problems for subclasses of chordal graphs: Minimum Fill-In, Interval Completion, Proper Interval Completion, 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

Minimum Fill-In: Inapproximability and Almost Tight Lower Bounds

“…Similar idea had actually been used by Kashiwabara and Fujisawa [22,23]. All hitherto known reductions on related problems, including [5,6,36], followed this idea, and used graph layout problems as the source problems. An important benefit of this approach is that they can be (usually in an effortless way) applied to related problems on (proper) interval graphs.…”

“…Yannakakis' reduction is far more powerful in this sense. It directly applies to interval graphs, unit interval graphs, and strongly chordal graph, while a slight modification works for trivially perfect graphs and threshold graphs (see, e.g., Bliznets et al [6] for details). Interval graphs are the intersection graphs of intervals on the real line.…”

“…Step 4 then blows up the graph size by another quadratic factor. Therefore, as already mentioned in [12,6] (without explanation), assuming ETH, one can only derive lower bounds of 2 o( 3 √ n) and 2 o( 6 √ k) · n O(1) from these reductions in their original form [14,37].…”

“…Bliznets et al [6] sedulously retraced the tortuous (and torturous) five-step reduction, and managed to decrease the size of the reduced instance by a stronger complexity conjecture and heavier constructions. First, they managed to improve step 3 such that it produces a graph of almost linear size, which allows them to derive improved but still weaker lower bounds of 2 O( √ n/ log c n) and 2 O( 4 √ k/ log c k) · n O(1) for some constant c. Then, to avoid the quadratic explosion in step 4, they had to introduce a new conjecture on the subexponential-time approximation hardness of the minimum bisection problem on d-regular graphs.…”

Exploring the Complexity of Layout Parameters in Tournaments and Semicomplete Digraphs

“…Another interesting direction is to investigate further the complexity of graph modification problems related to cutwidth: apply at most k modifications to the given semi-complete digraph in order to obtain a digraph of cutwidth at most c. Some immediate corollaries for vertex deletions are discussed in Section 5, but it is also interesting to look at the arc reversal variant, where the allowed modification is reversing an arc. For c = 0, this problem is equivalent to the Feedback Arc Set problem, which has been studied intensively in tournaments and semi-complete digraphs [1,4,10,11]. Further results on the vertex deletion and arc reversal problems related to cutwidth in semi-complete digraphs will be the topic of a future paper, currently under preparation.…”