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

2016
*
Algorithmica
*

Hence, using and improving

doi:10.1007/s00453-016-0249-1
fatcat:23bfo4vggrfmdfpeyqobwll6se
*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. ...##
###
A Deterministic Polynomial Kernel for Odd Cycle Transversal and Vertex Multiway Cut in Planar Graphs

2019
*
Symposium on Theoretical Aspects of Computer Science
*

On

doi:10.4230/lipics.stacs.2019.39
dblp:conf/stacs/JansenPL19
fatcat:u7f6rz4dl5dudfwsarvwga6le4
*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*. ...##
###
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices
[article]

2020
*
arXiv
*
pre-print

We also prove that there is an EPAS and a PSAKS

arXiv:1710.00668v4
fatcat:cgeptxhgobfg3ebkd3ul4so55e
*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. ...##
###
Kernelization Hardness of Connectivity Problems in d-Degenerate Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

We

doi:10.1007/978-3-642-16926-7_15
fatcat:irxbf2eapzaexa52bghnohykha
*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 ...##
###
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]

2017
*
arXiv
*
pre-print

In

arXiv:1306.6593v4
fatcat:oddhcwecevg5xhyo5mlqkzl3gm
*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. ...##
###
Kernelization hardness of connectivity problems in d-degenerate graphs

2012
*
Discrete Applied Mathematics
*

We

doi:10.1016/j.dam.2012.05.016
fatcat:xwx5p5lgvnebndqjkng56czfe4
*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 ...##
###
Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

2016
*
Dagstuhl Reports
*

This report documents

doi:10.4230/dagrep.6.5.94
dblp:journals/dagstuhl-reports/EricksonKMM16
fatcat:wasdfgivt5fqdppfxo3iqqs2ta
*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. ...##
###
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

2014
*
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
*

van (2014) Network sparsification

doi:10.1109/focs.2014.37
dblp:conf/focs/PilipczukPSL14
fatcat:33nagim3pvejpkiakhdive3zsy
*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 ...##
###
Parameterized Study of Steiner Tree on Unit Disk Graphs

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 ...

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

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 ...

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

2020
*
arXiv
*
pre-print

Parameterization and approximation are two popular ways

arXiv:2006.04411v1
fatcat:hjgu7f3s7zbydkcnioq3qlzgza
*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. ...##
###
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]

2018
*
arXiv
*
pre-print

On

arXiv:1810.01136v2
fatcat:hxolfduonvha3mlnhw2xiirurm
*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*. ...##
###
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

2020
*
Algorithms
*

Parameterization and approximation are two popular ways

doi:10.3390/a13060146
fatcat:2u2vv3uksfguvj6473t2gsq42a
*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. ...
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