# A Subexponential Parameterized Algorithm for Proper Interval Completion

@inproceedings{Bliznets2014ASP, title={A Subexponential Parameterized Algorithm for Proper Interval Completion}, author={Ivan A. Bliznets and F. Fomin and Marcin Pilipczuk and Michal Pilipczuk}, booktitle={Embedded Systems and Applications}, year={2014} }

In the Proper 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 a proper interval graph, i.e., a graph admitting an intersection model of equal-length intervals on a line. The study of Proper Interval Completion from the viewpoint of parameterized complexity has been initiated by Kaplan, Shamir and Tarjan [FOCS 1994; SIAM J. Comput. 1999], who showed an algorithm for the problem working in \(\mathcal{O}(16^k\cdot…

## 26 Citations

### Subexponential Parameterized Algorithm for Interval Completion

- Mathematics, Computer ScienceSODA
- 2016

The first subexponential parameterized algorithm solving Interval Completion in time kO(√k)nO(1) is given, adding IntervalCompletion to a very small list of parameterized graph modification problems solvable in subexp exponential time.

### Lower Bounds for the Parameterized Complexity of Minimum Fill-in and Other Completion Problems

- Computer Science, MathematicsSODA
- 2016

The second result proves that a significant improvement of any of these algorithms would lead to a surprising breakthrough in the design of approximation algorithms for MIN BISECTION, as well as improved, yet still not tight, lower bounds for FEEDBACK ARC SET in TOURNAMENTS.

### Exploring Subexponential Parameterized Complexity of Completion Problems

- MathematicsSTACS
- 2014

It is proved that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms.

### On the proper interval completion problem within some chordal subclasses

- Mathematics, Computer ScienceArXiv
- 2021

It is shown that the PIGcompletion problem within different subclasses of chordal graphs remains NP-complete even when restricted to split graphs, and an efficient algorithm is presented for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the Pig-com completion problem within this graph class.

### Paths to Trees and Cacti

- Mathematics, Computer ScienceCIAC
- 2017

The goal of designing polynomial kernels for trees with bounded number of leaves is considered and kernelization lower bounds are given for Bounded TC, Bounded OTC and Bounded CC by showing that unless NP coNP/poly the size of the kernel the authors obtain is optimal.

### Parameterized Graph Modification Algorithms

- Mathematics, Computer Science
- 2015

This thesis shows that editing towards trivially perfect graphs, threshold graphs, and chain graphs are all NP-complete, resolving 15 year old open questions and provides several new results in classical complexity, kernelization complexity, and subexponential parameterized complexity.

### On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

- MathematicsAPPROX-RANDOM
- 2020

This paper studies the problem of parameterized approximation of editing to a family of graphs by contracting edges by observing that the existing \textsf{W[2]-hardness} reduction can be adapted to show that there is no $F(k)$-\FPT-approximation algorithm for \textsc{Chordal Contraction}.

### Unit interval editing is fixed-parameter tractable

- Computer Science, MathematicsInf. Comput.
- 2015

### Rank Reduction of Directed Graphs by Vertex and Edge Deletions

- Mathematics, Computer ScienceLATIN
- 2016

The main structural result, which is the fulcrum of all the algorithmic results, is that for a fixed integer r the size of any “reduced graph” in \(\mathcal{F}_r\) is upper bounded by \(3^r\).

### FPT Algorithms for Plane Completion Problems

- Mathematics, Computer ScienceMFCS
- 2016

FPT algorithms that solve both problems in f (|E(∆)|) · |E(Γ)| 2 steps are given and it is shown that f(k)=2^{O(k*log(k))}.

## References

SHOWING 1-10 OF 36 REFERENCES

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- Mathematics, Computer ScienceSODA
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The first subexponential parameterized algorithm solving Interval Completion in time kO(√k)nO(1) is given, adding IntervalCompletion to a very small list of parameterized graph modification problems solvable in subexp exponential time.

### Exploring Subexponential Parameterized Complexity of Completion Problems

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It is proved that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms.

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This work presents a randomized subexponential time, polynomial space parameterized algorithm for the k -Weighted Feedback Arc Set in Tournaments (k -FAST ) problem and is the first non-trivial subexp exponential time parameterized algorithms outside the framework of bidimensionality.

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An algorithm with runtime O(k^{2k}n^3m) that performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search.

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The parameterized complexity of three NP-hard graph completion problems, motivated by molecular biology, is studied and it is shown that the parameterized version of the strongly chordal graph completion problem is FPT by giving an O(ck m log n)-time algorithm for it.

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This work gives the first subexponential parameterizedv algorithm solving Minimum Fill-in in time and substantially lowers the complexity of the problem.

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It is proved that a related problem, the so-called Bipartite Chain Deletion problem admits a kernel with O(k2) vertices, completing a previous result of Guo.

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We introduce a new framework for designing fixed-parameter algorithms with subexponential running time---2O(&kradic;) nO(1). Our results apply to a broad family of graph problems, called…

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