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… 
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26 Citations
Background Citations
13
Methods Citations
5

26 Citations

Subexponential Parameterized Algorithm for Interval Completion

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

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Exploring Subexponential Parameterized Complexity of Completion Problems

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

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Paths to Trees and Cacti

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

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.

Polynomial Kernelization for Removing Induced Claws and Diamonds

This work considers the parameterized complexity of the {claw,diamond}-free Edge Deletion problem, and proves that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time unless the Exponential Time Hypothesis fails.

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

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

  • Yixin Cao
  • Computer Science, Mathematics
    Inf. Comput.
  • 2015

Rank Reduction of Directed Graphs by Vertex and Edge Deletions

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 \(3^r\).

References

SHOWING 1-10 OF 36 REFERENCES

Subexponential Parameterized Algorithm for Interval Completion

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

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

Fast Fast

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.

Interval Completion Is Fixed Parameter Tractable

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.

Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs

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.

Subexponential parameterized algorithm for minimum fill-in

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

Polynomial kernels for Proper Interval Completion and related problems

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.

Faster Parameterized Algorithms for Deletion to Split Graphs

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.

Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs

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…

Subexponential parameterized algorithms on graphs of bounded-genus and H-minor-free graphs

A new framework for designing fixed-parameter algorithms with subexponential running time and a general approach for developing algorithms on H-minor-free graphs, based on structural results about Robertson & Seymour's graph-minors work are introduced.