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Hardness Results and Approximation Algorithms for the Minimum Dominating Tree Problem
[article]

2018
*
arXiv
*
pre-print

Given an undirected

arXiv:1802.04498v1
fatcat:yq3imtovtrep7hbvi75otz6l7m
*graph*G = (V, E)*with*some subsets of vertices called*groups*,*and*a*weight*function w:E →R, the*Group**Steiner**Tree*problem is to*find*a minimum*weight*sub-*tree*of G which contains at ... Given an undirected*graph*G = (V, E)*and*a*weight*function w:E →R, the Minimum Dominating*Tree*problem asks to*find*a minimum*weight*sub-*tree*of G, T = (U, F), such that every v ∈ V ∖ U is adjacent to ... Given an undirected*graph*G = (V, E)*with*some subsets of vertices called*groups*,*and*a*weight*function w : E → R, the*Group**Steiner**Tree*problem (GST) is to*find*a minimum*weight*sub-*tree*of G which contains ...##
###
On full Steiner trees in unit disk graphs

2015
*
Computational geometry
*

Given an

doi:10.1016/j.comgeo.2015.02.004
fatcat:2np2ox3zmncwznmypxm3bf6i5q
*edge*-*weighted**graph*G = (V, E)*and*a subset R of V , a*Steiner**tree*of G is a*tree*which spans all the vertices*in*R. ... A full*Steiner**tree*is a*Steiner**tree*which has all the vertices of R as its leaves. The full*Steiner**tree*problem is to*find*a full*Steiner**tree*of G*with*minimum*weight*. ... For a given*graph*G = (V, E), consider a complete*graph*G*with**vertex*set V . Define the*weight*of an*edge*(u, v)*in*G as the*weight*of the shortest path between u*and*v*in*G. ...##
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On the hardness of full Steiner tree problems

2015
*
Journal of Discrete Algorithms
*

A full

doi:10.1016/j.jda.2015.05.013
fatcat:rv72xuvpxrdcjgg7wbu5kogpe4
*Steiner**tree*is a*Steiner**tree**in*which each*vertex*of R is a leaf. The full*Steiner**tree*problem is to*find*a full*Steiner**tree**with*minimum*weight*. ... Given a*weighted**graph*G = (V, E)*and*a subset R of V , a*Steiner**tree**in*G is a*tree*which spans all vertices*in*R. The vertices*in*V \ R are called*Steiner*vertices. ... Given an*edge*or node-*weighted**graph*G, a root*vertex*r, a set G of*groups*, the goal is to*find*a minimum*weight*subgraph of G that contains two*edges*or*vertex*-disjoint paths from each*group**in*G to r ...##
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Online Node-Weighted Steiner Tree and Related Problems

2011
*
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
*

We obtain the first online algorithms for the node-

doi:10.1109/focs.2011.65
dblp:conf/focs/NaorPS11
fatcat:2lhdrbbql5h4nc5akxc7cju27y
*weighted**Steiner**tree*,*Steiner*forest*and**group**Steiner**tree*problems that achieve a poly-logarithmic competitive ratio. ... As further applications of our techniques, we also design polynomial-time online algorithms*with*poly-logarithmic competitive ratios for two fundamental network design problems*in**edge*-*weighted**graphs*: ... Partial support for this work was provided by ISF grant 954/11*and*BSF grant 2010426 for J. Naor,*and*NSF-STC Award 0939370 for D. Panigrahi. ...##
###
A PTAS for Node-Weighted Steiner Tree in Unit Disk Graphs
[chapter]

2009
*
Lecture Notes in Computer Science
*

Given a

doi:10.1007/978-3-642-02026-1_4
fatcat:hk3mwhpoajbt7dwbn4fe4uuaky
*graph*G = (V, E)*with*node*weight*function C : V → R +*and*a subset X of V , the node-*weighted**Steiner**tree*problem is to*find*a*Steiner**tree*for the set X such that its total*weight*is minimum ... As an application, we use node-*weighted**Steiner**tree*to solve the node-*weighted*connected dominating set problem*in*unit disk*graphs**and*obtain a (5+ε)-approximation algorithm. ... Given a*graph*G = (V, E)*with*node*weight*function C : V → R +*and*a subset X of V , which is denoted as the terminal set, the node-*weighted**Steiner**tree*problem is to*find*a*Steiner**tree*for the set X ...##
###
An O(nlogn) edge-based algorithm for obstacle-avoiding rectilinear steiner tree construction

2008
*
Proceedings of the 2008 international symposium on Physical design - ISPD '08
*

We adopt an

doi:10.1145/1353629.1353658
dblp:conf/ispd/LongZM08
fatcat:d2ra6cyxdvhxhalbxyixop6nr4
*edge*-based heuristic, which enables us to perform*both*local*and*global refinement, leading to*Steiner**trees**with*small lengths. The time complexity of our algorithm is O(nlogn). ...*In*this paper, we provide a new approach for rectilinear*Steiner**tree*construction*in*the presence of obstacles. ... Given a non-negative*weighted**graph*G, a directed sub-*graph*of G is called a terminal forest on G if 1) each*tree**in*the forest contains exactly one terminal*vertex**and*is rooted at this terminal,*and*...##
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Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

2016
*
Dagstuhl Reports
*

This report documents the program

doi:10.4230/dagrep.6.5.94
dblp:journals/dagstuhl-reports/EricksonKMM16
fatcat:wasdfgivt5fqdppfxo3iqqs2ta
*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. ...*With*that framework, we derive polynomial-time approximation schemes for the following problems*in*planar*graphs*or*graphs*of bounded genus:*edge*-*weighted**tree*cover*and*tour cover;*vertex*-*weighted*connected ...##
###
The relation of Connected Set Cover and Group Steiner Tree

2012
*
Theoretical Computer Science
*

We report that the Connected Set Cover (CSC) problem is just a special case of the

doi:10.1016/j.tcs.2012.02.035
fatcat:rjkf3ivm7fdw5owilh4osvwzzy
*Group**Steiner**Tree*(GST) problem. ... Based on that we obtain the first algorithm for CSC*with*polylogarithmic approximation guarantee as well as the first approximation algorithms for the*weighted*version of the problem*and*the version*with*...*Weighted*connected set cover problem is Ω(log 2−ϵ n)-hard, for all ϵ > 0. Conclusion*In*this paper, we found relation between two combinatorial problems, connected set cover*and**group**Steiner**tree*. ...##
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Minimum-cost heterogeneous node placement in wireless sensor networks

2019
*
IEEE Access
*

, produce

doi:10.1109/access.2019.2894117
fatcat:hbn4sumpi5h2lfkuk6fmrafvp4
*and*improve suboptimal solutions for large instances*with*100 000 vertices*and*1 000 000*edges*within 6 s. ... The objective is to minimize the sum of node production*and*placement costs*and*transmission outage probabilities*in*the routing*tree*. ... It is about*finding*the minimum-cost*tree**in*a*graph*to connect at least one*vertex**in*each*group*of vertices. ...##
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Towards Distributed 2-Approximation Steiner Minimal Trees in Billion-edge Graphs
[article]

2022
*
arXiv
*
pre-print

Furthermore, our distributed design exploits asynchronous processing

arXiv:2205.14503v1
fatcat:ifeouyji7rep5h7776ixnvjgqq
*and*a message prioritization scheme to accelerate convergence of distance computation,*and*harnesses*both**vertex**and**edge*centric processing ...*In*general, computing a*Steiner*minimal*tree*is NP-hard, but several polynomial-time algorithms have been designed*and*proven to yield*Steiner**trees*whose total*weight*is bounded within 2 times the*Steiner*... Funding from project LDRD #21-ERD-020 was used*in*this work. ...##
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Deploying Sensor Networks With Guaranteed Fault Tolerance

2010
*
IEEE/ACM Transactions on Networking
*

We have implemented greedy

doi:10.1109/tnet.2009.2024941
fatcat:imsothprlbfkrgcnvxwry3emnq
*and*distributed versions of this algorithm,*and*demonstrate*in*simulation that they produce high-quality placements for the additional sensors. ... Such a guarantee simultaneously provides fault tolerance against node failures*and*high overall network capacity (by the max-flow min-cut theorem). ... An important related problem is, given a*weighted*complete*graph**and*a number , to*find*minimum-*weight*-*vertex*-connected subgraph of . ...##
###
A greedy approximation algorithm for the group Steiner problem

2006
*
Discrete Applied Mathematics
*

*In*the

*group*

*Steiner*problem we are given an

*edge*-

*weighted*

*graph*G = (V , E, w)

*and*m subsets of vertices {g i } m i=1 . ... Each subset g i is called a

*group*

*and*the vertices

*in*i g i are called terminals. It is required to

*find*a minimum

*weight*

*tree*that contains at least one terminal from every

*group*. ... We thank the anonymous referees for extensive comments

*and*suggestions that improved the presentation of the paper. ...

##
###
Relay Placement for Higher Order Connectivity in Wireless Sensor Networks

2006
*
Proceedings IEEE INFOCOM 2006. 25TH IEEE International Conference on Computer Communications
*

*In*this paper we develop O(1)-approximation algorithms that

*find*close to optimal solutions

*in*time O((kn) 2 ) for achieving k-

*edge*connectivity of n nodes. ...

*Tree*

*in*d dimensions using Euclidean metrics. ... The algorithm constructs a complete

*graph*

*with*

*edge*

*weights*defined as

*in*Equation 1,

*and*

*find*a TSP tour on the

*graph*. The total

*weight*of the tour represents the number of relays required. ...

##
###
EBOARST: An Efficient Edge-Based Obstacle-Avoiding Rectilinear Steiner Tree Construction Algorithm

2008
*
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
*

Third, we present an

doi:10.1109/tcad.2008.2006098
fatcat:inxgblkcpbelpcw7s7g5cowcfa
*edge*-based heuristic, which enables us to perform*both*local*and*global refinements, leading to*Steiner**trees**with*small lengths. ...*In*this paper, we present EBOARST, an efficient four-step algorithm to construct a rectilinear obstacle-avoiding*Steiner**tree*for a given set of pins*and*a given set of rectilinear obstacles. ... Proof: For a given nonnegative*weighted**graph*G, we add one*vertex*v 0 to G,*and*add zero-*weighted**edge*(v 0 , u) to G for each terminal*vertex*u. We denote the newly obtained*graph*by G * . ...##
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A Graph Reduction Step Preserving Element-Connectivity and Applications
[article]

2009
*
arXiv
*
pre-print

Given an undirected

arXiv:0902.2795v1
fatcat:6t43enzedffzde5bqicnju3dfy
*graph*G=(V,E)*and*subset of terminals T ⊆ V, the element-connectivity of two terminals u,v ∈ T is the maximum number of u-v paths that are pairwise disjoint*in**both**edges**and*non-terminals ... Element-connectivity is more general than*edge*-connectivity*and*less general than*vertex*-connectivity. ... We thank Joseph Cheriyan for asking about planar packing of*Steiner**Trees*which inspired our work on that problem. ...
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