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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

We study extremal questions on induced matchings in several natural
graph classes. We argue that these questions should be asked for
twinless graphs, that is graphs not containing two vertices with
the same neighborhood. We show that planar twinless graphs always
contain an induced matching of size at least $n/40$ while there are
planar twinless graphs that do not contain an induced matching of
size $(n+10)/27$. We derive similar results for outerplanar graphs
and graphs of bounded genus. These extremal results can be applied
to the area of parameterized computation. For example, we show
that the induced matching problem on planar graphs has a kernel of
size at most $40k$ that is computable in linear time; this
significantly improves the results of Moser and Sikdar (2007). We
also show that we can decide in time $O(91^k + n)$ whether a planar
graph contains an induced matching of size at least $k$.

Iyad A. Kanj, Michael J. Pelsmajer, Ge Xia, and Marcus Schaefer. On the Induced Matching Problem. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 397-408, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{kanj_et_al:LIPIcs.STACS.2008.1361, author = {Kanj, Iyad A. and Pelsmajer, Michael J. and Xia, Ge and Schaefer, Marcus}, title = {{On the Induced Matching Problem}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {397--408}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1361}, URN = {urn:nbn:de:0030-drops-13618}, doi = {10.4230/LIPIcs.STACS.2008.1361}, annote = {Keywords: Induced matching, bounded genus graphs, parameterized algorithms, kernel} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

We consider the problem of constructing bounded-degree planar
geometric spanners of Euclidean and unit-disk graphs. It is well
known that the Delaunay subgraph is a planar geometric spanner with
stretch factor $C_{delapprox 2.42$; however, its degree may not be
bounded. Our first result is a very simple linear time algorithm
for constructing a subgraph of the Delaunay graph with stretch
factor $
ho =1+2pi(kcos{frac{pi{k)^{-1$ and degree bounded by
$k$, for any integer parameter $kgeq 14$. This result immediately
implies an algorithm for constructing a planar geometric spanner of
a Euclidean graph with stretch factor $
ho cdot C_{del$ and
degree bounded by $k$, for any integer parameter $kgeq 14$.
Moreover, the resulting spanner contains a Euclidean Minimum
Spanning Tree (EMST) as a subgraph. Our second contribution lies
in developing the structural results necessary to transfer our
analysis and algorithm from Euclidean graphs to unit disk graphs,
the usual model for wireless ad-hoc networks. We obtain a very
simple distributed, {em strictly-localized algorithm that, given a
unit disk graph embedded in the plane, constructs a geometric
spanner with the above stretch factor and degree bound, and also
containing an EMST as a subgraph. The obtained results
dramatically improve the previous results in all aspects, as shown
in the paper.

Ljubomir Perkovic and Iyad A. Kanj. On Geometric Spanners of Euclidean and Unit Disk Graphs. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 409-420, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{perkovic_et_al:LIPIcs.STACS.2008.1320, author = {Perkovic, Ljubomir and Kanj, Iyad A.}, title = {{On Geometric Spanners of Euclidean and Unit Disk Graphs}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {409--420}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1320}, URN = {urn:nbn:de:0030-drops-13200}, doi = {10.4230/LIPIcs.STACS.2008.1320}, annote = {Keywords: Geometric spanner, euclidean graph, unit disk graph, wireless ad-hoc networks} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

We study the parameterized complexity of a generalization of the coordinated motion planning problem on graphs, where the goal is to route a specified subset of a given set of k robots to their destinations with the aim of minimizing the total energy (i.e., the total length traveled). We develop novel techniques to push beyond previously-established results that were restricted to solid grids.
We design a fixed-parameter additive approximation algorithm for this problem parameterized by k alone. This result, which is of independent interest, allows us to prove the following two results pertaining to well-studied coordinated motion planning problems: (1) A fixed-parameter algorithm, parameterized by k, for routing a single robot to its destination while avoiding the other robots, which is related to the famous Rush-Hour Puzzle; and (2) a fixed-parameter algorithm, parameterized by k plus the treewidth of the input graph, for the standard Coordinated Motion Planning (CMP) problem in which we need to route all the k robots to their destinations. The latter of these results implies, among others, the fixed-parameter tractability of CMP parameterized by k on graphs of bounded outerplanarity, which include bounded-height subgrids.
We complement the above results with a lower bound which rules out the fixed-parameter tractability for CMP when parameterized by the total energy. This contrasts the recently-obtained tractability of the problem on solid grids under the same parameterization. As our final result, we strengthen the aforementioned fixed-parameter tractability to hold not only on solid grids but all graphs of bounded local treewidth - a class including, among others, all graphs of bounded genus.

Argyrios Deligkas, Eduard Eiben, Robert Ganian, Iyad Kanj, and M. S. Ramanujan. Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{deligkas_et_al:LIPIcs.ICALP.2024.53, author = {Deligkas, Argyrios and Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ramanujan, M. S.}, title = {{Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {53:1--53:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.53}, URN = {urn:nbn:de:0030-drops-201968}, doi = {10.4230/LIPIcs.ICALP.2024.53}, annote = {Keywords: coordinated motion planning, multi-agent path finding, parameterized complexity} }

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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

We study computationally-hard fundamental motion planning problems where the goal is to translate k axis-aligned rectangular robots from their initial positions to their final positions without collision, and with the minimum number of translation moves. Our aim is to understand the interplay between the number of robots and the geometric complexity of the input instance measured by the input size, which is the number of bits needed to encode the coordinates of the rectangles' vertices. We focus on axis-aligned translations, and more generally, translations restricted to a given set of directions, and we study the two settings where the robots move in the free plane, and where they are confined to a bounding box. We also consider two modes of motion: serial and parallel. We obtain fixed-parameter tractable (FPT) algorithms parameterized by k for all the settings under consideration.
In the case where the robots move serially (i.e., one in each time step) and axis-aligned, we prove a structural result stating that every problem instance admits an optimal solution in which the moves are along a grid, whose size is a function of k, that can be defined based on the input instance. This structural result implies that the problem is fixed-parameter tractable parameterized by k.
We also consider the case in which the robots move in parallel (i.e., multiple robots can move during the same time step), and which falls under the category of Coordinated Motion Planning problems. Our techniques for the axis-aligned motion here differ from those for the case of serial motion. We employ a search tree approach and perform a careful examination of the relative geometric positions of the robots that allow us to reduce the problem to FPT-many Linear Programming instances, thus obtaining an FPT algorithm.
Finally, we show that, when the robots move in the free plane, the FPT results for the serial motion case carry over to the case where the translations are restricted to any given set of directions.

Iyad Kanj and Salman Parsa. On the Parameterized Complexity of Motion Planning for Rectangular Robots. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 65:1-65:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kanj_et_al:LIPIcs.SoCG.2024.65, author = {Kanj, Iyad and Parsa, Salman}, title = {{On the Parameterized Complexity of Motion Planning for Rectangular Robots}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {65:1--65:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.65}, URN = {urn:nbn:de:0030-drops-200108}, doi = {10.4230/LIPIcs.SoCG.2024.65}, annote = {Keywords: motion planning of rectangular robots, coordinated motion planing of rectangular robots, parameterized complexity} }

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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Several works have recently investigated the parameterized complexity of data completion problems, motivated by their applications in machine learning, and clustering in particular. Interestingly, these problems can be equivalently formulated as classical graph problems on induced subgraphs of powers of partially-defined hypercubes.
In this paper, we follow up on this recent direction by investigating the Independent Set problem on this graph class, which has been studied in the data science setting under the name Diversity. We obtain a comprehensive picture of the problem’s parameterized complexity and establish its fixed-parameter tractability w.r.t. the solution size plus the power of the hypercube.
Given that several such FO-definable problems have been shown to be fixed-parameter tractable on the considered graph class, one may ask whether fixed-parameter tractability could be extended to capture all FO-definable problems. We answer this question in the negative by showing that FO model checking on induced subgraphs of hypercubes is as difficult as FO model checking on general graphs.

Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eiben_et_al:LIPIcs.IPEC.2023.16, author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ordyniak, Sebastian and Szeider, Stefan}, title = {{From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.16}, URN = {urn:nbn:de:0030-drops-194357}, doi = {10.4230/LIPIcs.IPEC.2023.16}, annote = {Keywords: Independent Set, Powers of Hypercubes, Diversity, Parameterized Complexity, Incomplete Data} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

In Coordinated Motion Planning (CMP), we are given a rectangular-grid on which k robots occupy k distinct starting gridpoints and need to reach k distinct destination gridpoints. In each time step, any robot may move to a neighboring gridpoint or stay in its current gridpoint, provided that it does not collide with other robots. The goal is to compute a schedule for moving the k robots to their destinations which minimizes a certain objective target - prominently the number of time steps in the schedule, i.e., the makespan, or the total length traveled by the robots. We refer to the problem arising from minimizing the former objective target as CMP-M and the latter as CMP-L. Both CMP-M and CMP-L are fundamental problems that were posed as the computational geometry challenge of SoCG 2021, and CMP also embodies the famous (n²-1)-puzzle as a special case.
In this paper, we settle the parameterized complexity of CMP-M and CMP-L with respect to their two most fundamental parameters: the number of robots, and the objective target. We develop a new approach to establish the fixed-parameter tractability of both problems under the former parameterization that relies on novel structural insights into optimal solutions to the problem. When parameterized by the objective target, we show that CMP-L remains fixed-parameter tractable while CMP-M becomes para-NP-hard. The latter result is noteworthy, not only because it improves the previously-known boundaries of intractability for the problem, but also because the underlying reduction allows us to establish - as a simpler case - the NP-hardness of the classical Vertex Disjoint and Edge Disjoint Paths problems with constant path-lengths on grids.

Eduard Eiben, Robert Ganian, and Iyad Kanj. The Parameterized Complexity of Coordinated Motion Planning. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eiben_et_al:LIPIcs.SoCG.2023.28, author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad}, title = {{The Parameterized Complexity of Coordinated Motion Planning}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {28:1--28:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.28}, URN = {urn:nbn:de:0030-drops-178784}, doi = {10.4230/LIPIcs.SoCG.2023.28}, annote = {Keywords: coordinated motion planning, multi-agent path finding, parameterized complexity, disjoint paths on grids} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We study two variants of the fundamental problem of finding a cluster in incomplete data. In the problems under consideration, we are given a multiset of incomplete d-dimensional vectors over the binary domain and integers k and r, and the goal is to complete the missing vector entries so that the multiset of complete vectors either contains (i) a cluster of k vectors of radius at most r, or (ii) a cluster of k vectors of diameter at most r. We give tight characterizations of the parameterized complexity of the problems under consideration with respect to the parameters k, r, and a third parameter that captures the missing vector entries.

Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. Finding a Cluster in Incomplete Data. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{eiben_et_al:LIPIcs.ESA.2022.47, author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ordyniak, Sebastian and Szeider, Stefan}, title = {{Finding a Cluster in Incomplete Data}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {47:1--47:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.47}, URN = {urn:nbn:de:0030-drops-169858}, doi = {10.4230/LIPIcs.ESA.2022.47}, annote = {Keywords: Parameterized complexity, incomplete data, clustering} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

We study fundamental point-line covering problems in computational geometry, in which the input is a set S of points in the plane. The first is the Rich Lines problem, which asks for the set of all lines that each covers at least λ points from S, for a given integer parameter λ ≥ 2; this problem subsumes the 3-Points-on-Line problem and the Exact Fitting problem, which - the latter - asks for a line containing the maximum number of points. The second is the NP-hard problem Line Cover, which asks for a set of k lines that cover the points of S, for a given parameter k ∈ ℕ. Both problems have been extensively studied. In particular, the Rich Lines problem is a fundamental problem whose solution serves as a building block for several algorithms in computational geometry.
For Rich Lines and Exact Fitting, we present a randomized Monte Carlo algorithm that achieves a lower running time than that of Guibas et al.’s algorithm [Computational Geometry 1996], for a wide range of the parameter λ. We derive lower-bound results showing that, for λ = Ω(√{n log n}), the upper bound on the running time of this randomized algorithm matches the lower bound that we derive on the time complexity of Rich Lines in the algebraic computation trees model.
For Line Cover, we present two kernelization algorithms: a randomized Monte Carlo algorithm and a deterministic algorithm. Both algorithms improve the running time of existing kernelization algorithms for Line Cover. We derive lower-bound results showing that the running time of the randomized algorithm we present comes close to the lower bound we derive on the time complexity of kernelization algorithms for Line Cover in the algebraic computation trees model.

Jianer Chen, Qin Huang, Iyad Kanj, and Ge Xia. Near-Optimal Algorithms for Point-Line Covering Problems. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chen_et_al:LIPIcs.STACS.2022.21, author = {Chen, Jianer and Huang, Qin and Kanj, Iyad and Xia, Ge}, title = {{Near-Optimal Algorithms for Point-Line Covering Problems}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {21:1--21:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.21}, URN = {urn:nbn:de:0030-drops-158312}, doi = {10.4230/LIPIcs.STACS.2022.21}, annote = {Keywords: line cover, rich lines, exact fitting, kernelization, randomized algorithms, complexity lower bounds, algebraic computation trees} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We present streaming algorithms for the graph k-matching problem in both the insert-only and dynamic models. Our algorithms, while keeping the space complexity matching the best known upper bound, have optimal or near-optimal update time, significantly improving on previous results. More specifically, for the insert-only streaming model, we present a one-pass randomized algorithm that runs in optimal 𝒪(k²) space and has optimal 𝒪(1) update time, and that, w.h.p. (with high probability), computes a maximum weighted k-matching of a weighted graph. Previously, the best upper bound on the update time was 𝒪(log k), which was achieved by a deterministic streaming algorithm that however only works for unweighted graphs [Stefan Fafianie and Stefan Kratsch, 2014]. For the dynamic streaming model, we present a one-pass randomized algorithm that, w.h.p., computes a maximum weighted k-matching of a weighted graph in Õ(Wk²) space and with Õ(1) update time, where W is the number of distinct edge weights. Again the update time of our algorithm improves the previous best upper bound Õ(k²) [Rajesh Chitnis et al., 2016]. Moreover, we prove that in the dynamic streaming model, any randomized streaming algorithm for the problem requires k²⋅ Ω(W(log W+1)) bits of space. Hence, both the space and update-time complexities achieved by our algorithm in the dynamic model are near-optimal. A streaming approximation algorithm for k-matching is also presented, whose space complexity matches the best known upper bound with a significantly improved update time.

Jianer Chen, Qin Huang, Iyad Kanj, Qian Li, and Ge Xia. Streaming Algorithms for Graph k-Matching with Optimal or Near-Optimal Update Time. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2021.48, author = {Chen, Jianer and Huang, Qin and Kanj, Iyad and Li, Qian and Xia, Ge}, title = {{Streaming Algorithms for Graph k-Matching with Optimal or Near-Optimal Update Time}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {48:1--48:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.48}, URN = {urn:nbn:de:0030-drops-154816}, doi = {10.4230/LIPIcs.ISAAC.2021.48}, annote = {Keywords: streaming algorithms, matching, parameterized algorithms, lower bounds} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem.
We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration.
We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise.

Dan Bergren, Eduard Eiben, Robert Ganian, and Iyad Kanj. On Covering Segments with Unit Intervals. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bergren_et_al:LIPIcs.STACS.2020.13, author = {Bergren, Dan and Eiben, Eduard and Ganian, Robert and Kanj, Iyad}, title = {{On Covering Segments with Unit Intervals}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.13}, URN = {urn:nbn:de:0030-drops-118741}, doi = {10.4230/LIPIcs.STACS.2020.13}, annote = {Keywords: Segment covering, unit intervals, NP-completeness, parameterized complexity} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G[A] and G[B] satisfy properties Pi_A and Pi_B, respectively. This so-called (Pi_A,Pi_B)-Recognition problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (Pi_A,Pi_B)-Recognition, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an "almost correct" bipartition (A',B'), and pushes appropriate vertices from A' to B' and vice versa to eventually arrive at a correct bipartition.
In this paper, we study whether (Pi_A,Pi_B)-Recognition problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where Pi_A is the set of P_3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and Pi_B is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP not subseteq coNP/poly, (Pi_A,Pi_B)-Recognition admits a polynomial kernel if and only if H contains a graph of order at most 2. In both the kernelization and the lower bound results, we make crucial use of the pushing process.

Iyad Kanj, Christian Komusiewicz, Manuel Sorge, and Erik Jan van Leeuwen. Solving Partition Problems Almost Always Requires Pushing Many Vertices Around. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kanj_et_al:LIPIcs.ESA.2018.51, author = {Kanj, Iyad and Komusiewicz, Christian and Sorge, Manuel and van Leeuwen, Erik Jan}, title = {{Solving Partition Problems Almost Always Requires Pushing Many Vertices Around}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.51}, URN = {urn:nbn:de:0030-drops-95140}, doi = {10.4230/LIPIcs.ESA.2018.51}, annote = {Keywords: Fixed-parameter algorithms, Kernelization, Vertex-partition problems, Reduction rules, Cross-composition} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? This is a fundamental problem that has undergone a tremendous amount of work by researchers in various areas, including computational geometry, graph theory, wireless computing, and motion planning. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be formulated and generalized into the following graph problem: Given a planar graph G whose vertices are colored by color sets, two designated vertices s, t in V(G), and k in N, is there an s-t path in G that uses at most k colors? If each obstacle is connected, the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph.
We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, among which a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms, as without it, the problem is W[SAT]-hard parameterized by k. Previous results only implied that the problem is W[2]-hard. A corollary of this result is that, unless W[2] = FPT, the problem cannot be approximated in FPT time to within a factor that is a function of k. By describing a generic plane embedding of the graph instances, we show that our hardness results translate to the geometric instances of the problem.
We then focus on graphs satisfying the color-connectivity property. By exploiting the planarity of the graph and the connectivity of the colors, we develop topological results that allow us to prove that, for any vertex v, there exists a set of paths whose cardinality is upper bounded by a function of k, that "represents" the valid s-t paths containing subsets of colors from v. We employ these structural results to design an FPT algorithm for the problem parameterized by both k and the treewidth of the graph, and extend this result further to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result generalizes and explains previous FPT results for various obstacle shapes, such as unit disks and fat regions.

Eduard Eiben and Iyad Kanj. How to Navigate Through Obstacles?. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eiben_et_al:LIPIcs.ICALP.2018.48, author = {Eiben, Eduard and Kanj, Iyad}, title = {{How to Navigate Through Obstacles?}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {48:1--48:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.48}, URN = {urn:nbn:de:0030-drops-90528}, doi = {10.4230/LIPIcs.ICALP.2018.48}, annote = {Keywords: parameterized complexity and algorithms, motion planning, barrier coverage, barrier resilience, colored path, minimum constraint removal, planar graphs} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Impagliazzo et al. proposed a framework, based on the logic fragment defining the complexity class SNP, to identify problems that are equivalent to k-CNF-Sat modulo subexponential-time reducibility (serf-reducibility). The subexponential-time solvability of any of these problems implies the failure of the Exponential Time Hypothesis (ETH). In this paper, we extend the framework of Impagliazzo et al., and identify a larger set of problems that are equivalent to k-CNF-Sat modulo serf-reducibility. We propose a complexity class, referred to as Linear Monadic NP, that consists of all problems expressible in existential monadic second order logic whose expressions have a linear measure in terms of a complexity parameter, which is usually the universe size of the problem.
This research direction can be traced back to Fagin's celebrated theorem stating that NP coincides with the class of problems expressible in existential second order logic. Monadic NP, a well-studied class in the literature, is the restriction of the aforementioned logic fragment to existential monadic second order logic. The proposed class Linear Monadic NP is then the restriction of Monadic NP to problems whose expressions have linear measure in the complexity parameter.
We show that Linear Monadic NP includes many natural complete problems such as the satisfiability of linear-size circuits, dominating set, independent dominating set, and perfect code. Therefore, for any of these problems, its subexponential-time solvability is equivalent to the failure of ETH. We prove, using logic games, that the aforementioned problems are inexpressible in the monadic fragment of SNP, and hence, are not captured by the framework of Impagliazzo et al. Finally, we show that Feedback Vertex Set is inexpressible in existential monadic second order logic, and hence is not in Linear Monadic NP, and investigate the existence of certain reductions between Feedback Vertex Set (and variants of it) and 3-CNF-Sat.

Robert Ganian, Ronald de Haan, Iyad Kanj, and Stefan Szeider. On Existential MSO and its Relation to ETH. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganian_et_al:LIPIcs.MFCS.2016.42, author = {Ganian, Robert and de Haan, Ronald and Kanj, Iyad and Szeider, Stefan}, title = {{On Existential MSO and its Relation to ETH}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {42:1--42:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.42}, URN = {urn:nbn:de:0030-drops-64556}, doi = {10.4230/LIPIcs.MFCS.2016.42}, annote = {Keywords: exponential time hypothesis (ETH), monadic second order logic, subexponential time complexity, serf-reducibility, logic games} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

We consider the recognition problem for two graph classes that generalize split and unipolar graphs, respectively.
First, we consider the recognizability of graphs that admit a monopolar partition: a partition of the vertex set into sets A,B such that G[A] is a disjoint union of cliques and G[B] an independent set. If in such a partition G[A] is a single clique, then G is a split graph. We show that in
O(2^k * k^3 * (|V(G)| + |E(G)|)) time we can decide whether G admits a monopolar partition
(A,B) where G[A] has at most k cliques. This generalizes the linear-time algorithm for recognizing split graphs corresponding to the case when k=1.
Second, we consider the recognizability of graphs that admit a 2-subcoloring: a partition of the vertex set into sets A,B such that each of G[A] and G[B] is a disjoint union of cliques. If in such a partition G[A] is a single clique, then G is a unipolar graph. We show that in
O(k^(2k+2) * (|V(G)|^2+|V(G)| * |E(G)|)) time we can decide whether G admits a
2-subcoloring (A,B) where G[A] has at most k cliques. This generalizes the polynomial-time algorithm for recognizing unipolar graphs corresponding to the case when k=1.
We also show that in O(4^k) time we can decide whether G admits a 2-subcoloring (A,B) where G[A] and G[B] have at most k cliques in total.
To obtain the first two results above, we formalize a technique, which we dub inductive recognition, that can
be viewed as an adaptation of iterative compression to recognition problems. We believe that the formalization
of this technique will prove useful in general for designing parameterized algorithms for recognition problems.
Finally, we show that, unless the Exponential Time Hypothesis fails, no subexponential-time algorithms for the
above recognition problems exist, and that, unless P=NP, no generic fixed-parameter algorithm exists for the
recognizability of graphs whose vertex set can be bipartitioned such that one part is a disjoint union of k
cliques.

Iyad Kanj, Christian Komusiewicz, Manuel Sorge, and Erik Jan van Leeuwen. Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kanj_et_al:LIPIcs.SWAT.2016.14, author = {Kanj, Iyad and Komusiewicz, Christian and Sorge, Manuel and Jan van Leeuwen, Erik}, title = {{Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.14}, URN = {urn:nbn:de:0030-drops-60360}, doi = {10.4230/LIPIcs.SWAT.2016.14}, annote = {Keywords: vertex-partition problems, monopolar graphs, subcolorings, split graphs, unipolar graphs, fixed-parameter algorithms} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p, q is realized as the line segment [pq], and is assigned a weight equal to the Euclidean distance |pq|. In this paper, we show how to construct in O(nlg{n}) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance.

Iyad Kanj, Ljubomir Perkovic, and Duru Türkoglu. Degree Four Plane Spanners: Simpler and Better. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kanj_et_al:LIPIcs.SoCG.2016.45, author = {Kanj, Iyad and Perkovic, Ljubomir and T\"{u}rkoglu, Duru}, title = {{Degree Four Plane Spanners: Simpler and Better}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {45:1--45:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.45}, URN = {urn:nbn:de:0030-drops-59376}, doi = {10.4230/LIPIcs.SoCG.2016.45}, annote = {Keywords: Nonnumerical Algorithms and Problems,Computational Geometry and Object Modeling} }

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Complete Volume

**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

LIPIcs, Volume 43, IPEC'15, Complete Volume

10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Proceedings{husfeldt_et_al:LIPIcs.IPEC.2015, title = {{LIPIcs, Volume 43, IPEC'15, Complete Volume}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015}, URN = {urn:nbn:de:0030-drops-56020}, doi = {10.4230/LIPIcs.IPEC.2015}, annote = {Keywords: Complexity Measures and Classes, Analysis of Algorithms and Problem Complexity, Discrete Mathematics} }

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Front Matter

**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Front Matter, Table of Contents, Preface, Program Committee, External Reviewers, List of Authors

10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. i-xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{husfeldt_et_al:LIPIcs.IPEC.2015.i, author = {Husfeldt, Thore and Kanj, Iyad}, title = {{Front Matter, Table of Contents, Preface, Program Committee, External Reviewers, List of Authors}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {i--xiv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.i}, URN = {urn:nbn:de:0030-drops-55676}, doi = {10.4230/LIPIcs.IPEC.2015.i}, annote = {Keywords: IPEC} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Let T be a triangulation of a set P of n points in the plane, and let e be an edge shared by two triangles in T such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q to obtain a new triangulation of P from T. The flip distance between two triangulations of P is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of P is k, for some given k \in \mathbb{N}. It is a fundamental and a challenging problem.
In this paper we present an algorithm for the Flip Distance problem that
runs in time O(n + k \cdot c^{k}), for a constant c \leq 2 \cdot
14^11, which implies that the problem is fixed-parameter tractable. The
NP-hardness reduction for the Flip Distance problem given by Lubiw
and Pathak can be used to show that, unless the exponential-time hypothesis (ETH) fails, the Flip Distance problem cannot be solved in time O^*(2^o(k)). Therefore, one cannot expect an asymptotic improvement in the exponent of the running time of our algorithm.

Iyad Kanj and Ge Xia. Flip Distance Is in FPT Time O(n+ k * c^k). In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 500-512, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kanj_et_al:LIPIcs.STACS.2015.500, author = {Kanj, Iyad and Xia, Ge}, title = {{Flip Distance Is in FPT Time O(n+ k * c^k)}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {500--512}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.500}, URN = {urn:nbn:de:0030-drops-49371}, doi = {10.4230/LIPIcs.STACS.2015.500}, annote = {Keywords: triangulations, flip distance, parameterized algorithms} }

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