# 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}
}
• Published in
Embedded Systems and…
13 February 2014
• Mathematics, Computer Science
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 • Mathematics, Computer Science SODA • 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. • Computer Science, Mathematics SODA • 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. • Mathematics STACS • 2014 It is proved that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms. • Mathematics, Computer Science ArXiv • 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. • Mathematics, Computer Science CIAC • 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. 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. • Mathematics APPROX-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}. • Mathematics, Computer Science LATIN • 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$$.
• Mathematics, Computer Science
MFCS
• 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

• Mathematics, Computer Science
SODA
• 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.
• Mathematics
STACS
• 2014
It is proved that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms.
• Mathematics, Computer Science
ICALP
• 2009
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.
• Computer Science
SIAM J. Comput.
• 2009
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.
• Computer Science, Mathematics
SIAM J. Comput.
• 1999
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.
• Mathematics
SODA
• 2012
This work gives the first subexponential parameterizedv algorithm solving Minimum Fill-in in time and substantially lowers the complexity of the problem.
• Mathematics
FCT
• 2011
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.
• Mathematics, Computer Science
Algorithmica
• 2013
A systematic study of parameterized complexity of the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split and an efficient fixed-parameter algorithms and polynomial sized kernels are given.
• Mathematics
JACM
• 2005
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