Unit Interval Editing is Fixed-Parameter Tractable

Abstract

Given a graph G and integers k 1 , k 2 , and k 3 , the unit interval editing problem asks whether G can be transformed into a unit interval graph by at most k 1 vertex deletions, k 2 edge deletions, and k 3 edge additions. We give an algorithm solving this problem in time 2 O(k log k) · (n + m), where k := k 1 + k 2 + k 3 , and n, m denote respectively the numbers of vertices and edges of G. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations.Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time O(4 k · (n + m)). Another result is an O(6 k · (n + m))-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time O(6 k · n 6 ).

Quick but Odd Growth of Cacti

“…These problems have led to identification of several new techniques and ideas in the field. Recent years have seen a plethora of results around vertex and edge deletion problems, in the domain of parameterized complexity [3,4,[8][9][10][11][12]. In this paper, we continue this line of research and study two vertex deletion problems.…”

Unit interval vertex deletion: Fewer vertices are relevant

“…We have been using the approximation algorithm [3] as a black box for furnishing the modulator. To have a better analysis, we may have to unwrap the black box and see a bit of how it works.…”

“…We can redo the second phase in O(m) time to produce an approximation solution for the new graph. 3 Therefore, we can apply each reduction rule and presently recover the modulator in O(m) time. On the other hand, since each application of a reduction rule deletes at least one vertex from the graph, they can be applied at most n times.…”

“…As a final remark, properties of the approximation algorithm [3] may be further exploited to sharpen the analysis of the kernel size of our kernelization algorithm. But it would very unlikely lead to one with o(k 2 ) vertices.…”

“…One can show that it can be easily adapted to contracting all but 6 cliques in a sequence of cliques in K that are nonadjacent to M. Moreover, all the cliques can be handled in one run, in linear time. To prove these facts, however, we need to revisit the approximation algorithm[3] with all the details, which we omit to not blur the focus of the current paper.…”