scholarly journals The Clustered Selected-Internal Steiner Tree Problem

2021 ◽  
pp. 1-12
Author(s):  
Yen Hung Chen
Keyword(s):  
Approximation Algorithm ◽  
Complete Graph ◽  
Steiner Tree ◽  
Minimum Cost ◽  
Performance Ratio ◽  
Steiner Tree Problem ◽  
The Cost

Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets [Formula: see text] and [Formula: see text], a partition [Formula: see text] of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], a clustered Steiner tree is a tree [Formula: see text] of [Formula: see text] that spans all vertices in [Formula: see text] such that [Formula: see text] can be cut into [Formula: see text] subtrees [Formula: see text] by removing [Formula: see text] edges and each subtree [Formula: see text] spans all vertices in [Formula: see text], [Formula: see text]. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of [Formula: see text] is a clustered Steiner tree for [Formula: see text] if all vertices in [Formula: see text] are internal vertices of [Formula: see text]. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree [Formula: see text] for [Formula: see text] and [Formula: see text] in [Formula: see text] with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio [Formula: see text] for the clustered selected-internal Steiner tree problem, where [Formula: see text] is the best-known performance ratio for the Steiner tree problem.

scholarly journals Algorithms for the power-p Steiner tree problem in the Euclidean plane

2018 ◽  
Vol 25 (4) ◽  
pp. 28
Author(s):  
Christina Burt ◽  
Alysson Costa ◽  
Charl Ras
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Valid Inequalities ◽  
Minimum Cost ◽  
Performance Ratio ◽  
Mixed Integer ◽  
Steiner Tree Problem ◽  
Steiner Points ◽  
Tree Construction ◽  
The Cost

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case.

MULTICASTING AND BROADCASTING IN UNDIRECTED WDM NETWORKS AND QoS EXTENTIONS OF MULTICASTING

2004 ◽  
Vol 15 (01) ◽  
pp. 187-203
Author(s):  
YINLONG XU ◽  
LI LIN ◽  
GUOLIANG CHEN ◽  
YINGYU WAN ◽  
WEIJUN GUO
Keyword(s):  
Steiner Tree ◽  
Constant Factor ◽  
Wdm Networks ◽  
Approximate Algorithm ◽  
Performance Ratio ◽  
Steiner Tree Problem ◽  
Log P ◽  
Auxiliary Graph ◽  
Group Steiner Tree ◽  
The Cost

This paper addresses multicasting and broadcasting in undirected WDM networks and QoS extensions of multicasting. It is given an undirected network G=(V, E), with Λ is the set of the available wavelengths in G, and associated with each edge, there is a subset of wavelengths on it. For a multicast request r=(s, D) with a source s and a set D of destinations, it is to find a tree rooted at s including all nodes in D such that the cost of the tree is minimized in terms of the cost of wavelength conversion at nodes and the cost of using wavelength on edges. This paper proves that multicasting in this model of networks is NP-Hard and cannot be approximated within a constant factor, unless P=NP. Furthermore, an auxiliary graph is constructed for the original WDM network, the multicasting is reduced to a group Steiner tree problem on the auxiliary graph and an approximate algorithm based on the group Steiner tree algorithm proposed by M. Charikar et al. with performance ratio of O( log 2(nk) log log (nk) log p) is provided, where k=|Λ| and p=|D∪{s}|. At last, some QoS extensions of multicasting are discussed.

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