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Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices

Ondřej Suchý
2016 Algorithmica  
Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in planar graphs for the parameterization by the number k of Steiner vertices in the solution.  ...  Recently, Pilipczuk, Pilipczuk, Sankowski, and van Leeuwen [FOCS 2014] gave a polynomial kernel for Steiner Tree in planar graphs, when parameterized by |T | + k, the total number of vertices in the constructed  ...  Conclusion and Future Directions We presented the first polynomial kernel for Planar Steiner Tree with respect to the number k of Steiner points in the solution.  ... 
doi:10.1007/s00453-016-0249-1 fatcat:23bfo4vggrfmdfpeyqobwll6se

A Deterministic Polynomial Kernel for Odd Cycle Transversal and Vertex Multiway Cut in Planar Graphs

Bart M. P. Jansen, Marcin Pilipczuk, Erik Jan Van Leeuwen, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science  
On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014).  ...  of) Steiner trees after adding all edges between pairs of vertices that lie on a common face.  ...  To this end, we adapt the rules that Suchý [28] developed for Plane Steiner Tree parameterized by the number of Steiner vertices of the solution tree.  ... 
doi:10.4230/lipics.stacs.2019.39 dblp:conf/stacs/JansenPL19 fatcat:u7f6rz4dl5dudfwsarvwga6le4

Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices [article]

Pavel Dvořák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masařík, Tomáš Toufar, Pavel Veselý
2020 arXiv   pre-print
We also prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter  ...  In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution.  ...  We would like to thank Michael Lampis and Édouard Bonnet for helpful discussions on the problem.  ... 
arXiv:1710.00668v4 fatcat:cgeptxhgobfg3ebkd3ul4so55e

Kernelization Hardness of Connectivity Problems in d-Degenerate Graphs [chapter]

Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, Jakub Onufry Wojtaszczyk
2010 Lecture Notes in Computer Science  
We extend our analysis by showing that, unless P H = Σ 3 p , there do not exist polynomial kernels for STEINER TREE, CONNECTED FEEDBACK VERTEX SET and CONNECTED ODD CYCLE TRANSVERSAL in d-degenerate graphs  ...  Our technique yields also an alternative proof that, under the same complexity assumption, STEINER TREE does not admit a polynomial kernel.  ...  In particular, such techniques may lead to a negative result for the question of existence of a polynomial kernel for PLANAR STEINER TREE, which today is a major open problem in kernelization and is not  ... 
doi:10.1007/978-3-642-16926-7_15 fatcat:irxbf2eapzaexa52bghnohykha

Page 3688 of Mathematical Reviews Vol. , Issue 82i [page]

1982 Mathematical Reviews  
In an edge-weighted graph G, a tree S in G which interconnects vertices in BC V(G) is called a Steiner tree for B.  ...  The author proves that the worst case bound for the ratio of the lengths of the Steiner minimal tree and minimal spanning tree tends to 2. This result, however, is a special case of a result of E. F.  ... 

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs [article]

Marcin Pilipczuk, Michał Pilipczuk, Piotr Sankowski, Erik Jan van Leeuwen
2017 arXiv   pre-print
In the language of parameterized complexity, these results imply the first polynomial kernels for Steiner Tree and Steiner Forest on planar and bounded-genus graphs (parameterized by the size of the tree  ...  E(G) of size polynomial in g and k that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of f.  ...  Moreover, we acknowledge the discussions with Daniel Lokshtanov that lead to the discovery that the NP-hardness proof for Steiner Forest on planar graphs of treewidth 3 of Bateni et al.  ... 
arXiv:1306.6593v4 fatcat:oddhcwecevg5xhyo5mlqkzl3gm

Kernelization hardness of connectivity problems in d-degenerate graphs

Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, Jakub Onufry Wojtaszczyk
2012 Discrete Applied Mathematics  
We extend our analysis by showing that, unless NP ⊆ coNP/poly, there do not exist polynomial kernels for Steiner Tree, Connected Feedback Vertex Set and Connected Odd Cycle Transversal in d-degenerate  ...  Our technique also yields an alternative proof that, under the same complexity assumption, Steiner Tree does not admit a polynomial kernel.  ...  In particular, such techniques may lead to a negative result for the question of existence of a polynomial kernel for Planar Steiner Tree, which today is a major open problem in kernelization and is not  ... 
doi:10.1016/j.dam.2012.05.016 fatcat:xwx5p5lgvnebndqjkng56czfe4

Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

Jeff Erickson, Philip N. Klein, Dániel Marx, Claire Mathieu, Marc Herbstritt
2016 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 16221 "Algorithms for Optimization Problems in Planar Graphs". The seminar was held from May 29 to June 3, 2016.  ...  This report contains abstracts for the recent developments in planar graph algorithms discussed during the seminar as well as summaries of open problems in this area of research.  ...  For this proof we introduce the notion of primal-dual graph and we extend the planar linkage theorem for this type of graphs.  ... 
doi:10.4230/dagrep.6.5.94 dblp:journals/dagstuhl-reports/EricksonKMM16 fatcat:wasdfgivt5fqdppfxo3iqqs2ta

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

Marcin Pilipczuk, Michal Pilipczuk, Piotr Sankowski, Erik Jan van Leeuwen
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
van (2014) Network sparsification for Steiner problems on planar and bounded-genus graphs.  ...  Moreover, we acknowledge the discussions with Daniel Lokshtanov that lead to the discovery that the NP-hardness proof for Steiner Forest on planar graphs of treewidth 3 of Bateni et al.  ...  Acknowledgements We thank Daniel Lokshtanov and Saket Saurabh for showing us the application of Baker's approach to Planar Multiway Cut (Theorem 13.1) and for allowing us to include the proof in this paper  ... 
doi:10.1109/focs.2014.37 dblp:conf/focs/PilipczukPSL14 fatcat:33nagim3pvejpkiakhdive3zsy

Parameterized Study of Steiner Tree on Unit Disk Graphs

Sujoy Bhore, Paz Carmi, Sudeshna Kolay, Meirav Zehavi, Susanne Albers
2020 Scandinavian Workshop on Algorithm Theory  
The vertices of R are referred to as terminals and the vertices of V(G)⧵ R as Steiner vertices. First, we show that the problem is NP-hard.  ...  We mention that the algorithmic results can be made to work for Steiner Tree on disk graphs with bounded aspect ratio.  ...  Our algorithm is based on the idea that for an instance (G, R, t, k), in order to determine the existence of a Steiner tree we can first find spanning trees for all components of G[R] and extend these  ... 
doi:10.4230/lipics.swat.2020.13 dblp:conf/swat/BhoreCKZ20 fatcat:5hl32ecyofhj3aidc73jspmn2e

Parameterized Study of Steiner Tree on Unit Disk Graphs [article]

Sujoy Bhore, Paz Carmi, Sudeshna Kolay, Meirav Zehavi
2020 arXiv   pre-print
The vertices of R are referred to as terminals and the vertices of V(G)∖ R as Steiner vertices. First, we show that the problem is NP-Hard.  ...  We mention that the algorithmic results can be made to work for the Steiner Tree on disk graphs with bounded aspect ratio.  ...  Our algorithm is based on the idea that for an instance (G, R, t, k), in order to determine the existence of a Steiner tree we can first find spanning trees for all components of G[R] and extend these  ... 
arXiv:2004.09220v1 fatcat:quagp4jzojbj5divftlmz2plke

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms [article]

Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, Pasin Manurangsi
2020 arXiv   pre-print
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results.  ...  We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
arXiv:2006.04411v1 fatcat:hjgu7f3s7zbydkcnioq3qlzgza

Page 2452 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews  
The extent e(G) of a graph G is the minimal number 1 +27 such that there exists a tree T in G which is at distance at most n from all the vertices of G.  ...  Summary: “For a nonempty subset S of vertices of a k-connected graph G and for 1 <i<k, the Steiner i-distance d;(S) of S is the minimum size among all i-connected subgraphs containing S.  ... 

A deterministic polynomial kernel for Odd Cycle Transversal and Vertex Multiway Cut in planar graphs [article]

Bart M.P. Jansen and Marcin Pilipczuk and Erik Jan van Leeuwen
2018 arXiv   pre-print
On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014).  ...  of) Steiner trees after adding all edges between pairs of vertices that lie on a common face.  ...  To this end, we adapt the rules that Suchý [30] developed for Plane Steiner Tree parameterized by the number of Steiner vertices of the solution tree.  ... 
arXiv:1810.01136v2 fatcat:hxolfduonvha3mlnhw2xiirurm

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, Pasin Manurangsi
2020 Algorithms  
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results.  ...  We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/a13060146 fatcat:2u2vv3uksfguvj6473t2gsq42a
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