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

2018 arXiv   pre-print
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 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

Ahmad Biniaz, Anil Maheshwari, Michiel Smid
2015 Computational geometry
Given an 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.  ...

### On the hardness of full Steiner tree problems

Ahmad Biniaz, Anil Maheshwari, Michiel Smid
2015 Journal of Discrete Algorithms
A full 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  ...

### Online Node-Weighted Steiner Tree and Related Problems

J. Naor, D. Panigrahi, M. Singh
2011 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
We obtain the first online algorithms for the node-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]

Xianyue Li, Xiao-Hua Xu, Feng Zou, Hongwei Du, Pengjun Wan, Yuexuan Wang, Weili Wu
2009 Lecture Notes in Computer Science
Given a 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

Jieyi Long, Hai Zhou, Seda Ogrenci Memik
2008 Proceedings of the 2008 international symposium on Physical design - ISPD '08
We adopt an 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  ...

### 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.  ...  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

Khaled Elbassioni, Slobodan Jelić, Domagoj Matijević
2012 Theoretical Computer Science
We report that the Connected Set Cover (CSC) problem is just a special case of the 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.  ...

### Minimum-cost heterogeneous node placement in wireless sensor networks

Yahui Sun, Saman Halgamuge
2019 IEEE Access
, produce 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.  ...

### Towards Distributed 2-Approximation Steiner Minimal Trees in Billion-edge Graphs [article]

Tahsin Reza, Geoffrey Sanders, Roger Pearce
2022 arXiv   pre-print
Furthermore, our distributed design exploits asynchronous processing 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.  ...

### Deploying Sensor Networks With Guaranteed Fault Tolerance

J.L. Bredin, E.D. Demaine, M.T. Hajiaghayi, D. Rus
2010 IEEE/ACM Transactions on Networking
We have implemented greedy 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

Chandra Chekuri, Guy Even, Guy Kortsarz
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

A. Kashyap, S. Khuller, M. Shayman
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

Jieyi Long, Hai Zhou, Seda Ogrenci Memik
2008 IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Third, we present an 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 * .  ...

### A Graph Reduction Step Preserving Element-Connectivity and Applications [article]

Chandra Chekuri, Nitish Korula
2009 arXiv   pre-print
Given an undirected 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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