Bottleneck Steiner Tree Problem Research Articles

A bi-objective network design approach for discovering functional modules linking Golgi apparatus fragmentation and neuronal death

Experimental records show the existence of a biological linkage between neuronal death and Golgi apparatus fragmentation. The comprehension of such linkage should help to understand the dynamics undergoing neurological damage caused by diseases such as Alzheimer’s disease or amyotrophic lateral sclerosis. In this paper, the bi-objective minimum cardinality bottleneck Steiner tree problem along with an ad-hoc exact algorithm are proposed to study such phenomena. The proposed algorithm is based on integer programming and the so-called $$\epsilon $$ -constraint method. A key feature of the devised approach is that it allows an efficient integer programming formulation of the problem. The obtained results show that it is possible to obtain additional evidence supporting the hypothesis that alterations of the Golgi apparatus structure and neuronal death interact through the biological mechanisms underlying the outbreak and progression of neurodegenerative diseases. Moreover, the function of cellular response to stress as a biological linkage between these phenomena is also further investigated. Complementary, computational results on a synthetic dataset are also provided with the aim of reporting the performance of the proposed algorithm.

Determining Sensor Locations in Wireless Sensor Networks

Network lifetime plays an important role in the design of wireless sensor networks. This paper studies the problem of prolonging the wireless sensor network's lifetime, through introducing additional sensors at proper locations to achieve the goal of minimizing the length of the longest edge in the network. The problem is in fact the bottleneck Steiner tree problem, trying to find a Steiner tree minimizing the length of the longest edges for the given n terminals in the Euclidean plane by introducing at most k Steiner points. A restricted bottleneck Steiner tree problem is studied in this paper, which requires that only degree ≥3 Steiner points are not allowed to be adjacent in the optimal solution. We show that the restricted problem is MAX-SNP hard and cannot be approximated within performance ratio [Formula: see text] in polynomial time unless P = NP; we first propose a polynomial time [Formula: see text]-approximation algorithm and then improve the ratio to [Formula: see text] for any given [Formula: see text], by presenting a polynomial time randomized approximation algorithm, which is almost optimal to the restricted problem.

Survivable minimum bottleneck networks

We show that the survivable bottleneck Steiner tree problem in normed planes can be solved in polynomial time when the number of Steiner points is constant. This is a fundamental problem in wireless ad-hoc network design where the objective is to design networks with efficient routing topologies. Our result holds for a general definition of survivability and for any norm whose ball is specified by a piecewise algebraic curve of bounded degree and with a bounded number of pieces. In particular, under the Euclidean and rectilinear norms our algorithm constructs an optimal solution in O(n2k+3log⁡n) steps, where n is the number of terminals and k is the number of Steiner points. Our algorithm is based on the construction of generalised Voronoi diagrams and relative neighbourhood graphs.

Fast Approximation Algorithm for Restricted Euclidean Bottleneck Steiner Tree Problem

Bottleneck Steiner tree problem asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. The problem has applications in the design of wireless communication networks. In this paper we study a restricted version of the bottleneck Steiner tree problem in the Euclidean plane which requires that only degree-2 Steiner points are possibly adjacent in the optimal solution. We first show that the problem is NP-hard and cannot be approximated within unless P=NP, and provide a fast polynomial time deterministic approximation algorithm with performance ratio .

Nearly Optimal Solution for Restricted Euclidean Bottleneck Steiner Tree Problem

A variation of the traditional Steiner tree problem, the bottleneck Steiner tree problem is considered in this paper, which asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. The problem has applications in the design of WDM optical networks, design of wireless communication networks and reconstruction of phylogenetic tree in biology. We study a restricted version of the bottleneck Steiner tree problem in the Euclidean plane which requires that only degree-2 Steiner points are possibly adjacent in the optimal solution. The problem is known to be MAX-SNP hard and cannot be approximated within unless P=NP, we propose a nearly optimal randomized polynomial time approximation algorithm with performance ratio +e, where e is a positive number.

Relay augmentation for lifetime extension of wireless sensor networks

The authors propose a novel relay augmentation strategy for extending the lifetime of a certain class of wireless sensor networks. In this class sensors are located at fixed and pre-determined positions and all communication takes place through multi-hop paths in a fixed routing tree rooted at the base station. It is assumed that no accumulation of data takes place along the communication paths and that there is no restriction on where additional relays may be located. Under these assumptions the optimal extension of network lifetime is modelled as the Euclidean k -bottleneck Steiner tree problem. Only two approximation algorithms for this NP-hard problem exist in the literature: a minimum spanning tree heuristic (MSTH) with performance ratio 2, and a probabilistic 3-regular hypergraph heuristic (3RHH) with performance ratio √3 + ɛ. The authors present a new iterative heuristic that incorporates MSTH and show through simulation that their algorithm performs better than MSTH in extending lifetime, and outperforms 3RHH in terms of efficiency.

Generalised k-Steiner Tree Problems in Normed Planes

The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within $O(n^2)$ time. In this paper we generalise their approach in order to solve the $k$-Steiner tree problem, in which the Steiner minimum tree may contain up to $k$ Steiner points for a given constant $k$. We also extend their approach further to encompass other normed planes, and to solve a much wider class of problems, including the $k$-bottleneck Steiner tree problem and other generalised $k$-Steiner tree problems. We show that, for any fixed $k$, such problems can be solved in $O(n^{2k})$ time.

Exact Algorithms for the Bottleneck Steiner Tree Problem

Given n points, called terminals, in the plane ℝ2 and a positive integer k, the bottleneck Steiner tree problem is to find k Steiner points from ℝ2 and a spanning tree on the n+k points that minimi...

On exact solutions to the Euclidean bottleneck Steiner tree problem

We study the Euclidean bottleneck Steiner tree problem: given a set P of n points in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edge in the tree is minimized. This problem is known to be NP-hard even to approximate within ratio 2 and there was no known exact algorithm even for k = 1 prior to this work. In this paper, we focus on finding exact solutions to the problem for a small constant k. Based on geometric properties of optimal location of Steiner points, we present an optimal Θ ( n log n ) -time exact algorithm for k = 1 and an O ( n 2 ) -time algorithm for k = 2 . Also, we present an optimal Θ ( n log n ) -time exact algorithm for any constant k for a special case where there is no edge between Steiner points.

Improved polynomial algorithms for robust bottleneck problems with interval data

To model uncertainties in a problem one can provide intervals of uncertainty specifying a range for each uncertain parameter value. To hedge against ‘worst-case scenarios’, i.e., most unwelcome realizations of the uncertain parameters after a solution has been determined, the minmax-regret criterion has been adopted in robust optimization. Within this field, we consider bottleneck problems with one or more uncertain parameter functions. We apply a known polynomial time solution scheme for a number of specific problems of this type with one uncertain parameter function. This leads to improved algorithms of reduced complexity, e.g., for the bottleneck Steiner tree problem and the bottleneck assignment problem. Further, we formulate a framework to solve single machine scheduling problems in polynomial time under the maximum tardiness criterion, with up to three uncertain parameter functions.

Optimal Relay Location for Resource-limited Energy-efficient Wireless Communication

In the design of wireless networks, techniques for improving energy efficiency and extending network lifetime have great importance, particularly for defense and civil/rescue applications where resupplying transmitters with new batteries is not feasible. In this paper we study a method for improving the lifetime of wireless networks by minimizing the length of the longest edge in the interconnecting tree by deploying additional relay nodes at specific locations. This optimization problem, known as the Bottleneck Steiner Tree Problem (BSTP), asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. We present a ratio- \(\sqrt{3}+\epsilon\) polynomial time approximation algorithm for BSTP, where \(\epsilon\) is an arbitrary positive number.

Variations of the prize‐collecting Steiner tree problem

AbstractThe prize‐collecting Steiner tree problem is well known to be NP‐hard. We consider seven variations of this problem generalizing several well‐studied bottleneck and minsum problems with feasible solutions as trees of a graph. Four of these problems are shown to be solvable in O(m+n log n) time and the remaining are shown to be NP‐hard where n is the number of nodes and m is the number of edges in the underlying graph. For one of these polynomially solvable cases, we also provide an O(m) algorithm generalizing and unifying known linear time algorithms for the bottleneck spanning tree problem, bottleneck s−t path problem, and bottleneck Steiner tree problem. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(4), 199–205 2006

Approximation algorithm for bottleneck Steiner tree problem in the Euclidean plane

A special case of the bottleneck Steiner tree problem in the Euclidean plane was considered in this paper. The problem has applications in the design of wireless communication networks, multifacility location, VLSI routing and network routing. For the special case which requires that there should be no edge connecting any two Steiner points in the optimal solution, a 3-restricted Steiner tree can be found indicating the existence of the performance ratio √2. In this paper, the special case of the problem is proved to be NP-hard and cannot be approximated within ratio √2. First a simple polynomial time approximation algorithm with performance ratio √3 is presented. Then based on this algorithm and the existence of the 3-restricted Steiner tree, a polynomial time approximation algorithm with performance ratio--√2 + e is proposed, for any e > 0.

A bicriterion Steiner tree problem on graph

This paper presents a formulation of bicriterion Steiner tree problem which is stated as a task of finding a Steiner tree with maximal capacity and minimal length. It is considered as a lexicographic multicriteria problem. This means that the bottleneck Steiner tree problem is solved first. After that, the next optimization problem is stated as a classical minimums Steiner tree problem under the constraint on capacity of the tree. The paper also presents some computational experiments with the multicriteria problem.

An approximation algorithm for a bottleneck k-Steiner tree problem in the Euclidean plane

We study a bottleneck Steiner tree problem: given a set P={ p 1, p 2,…, p n } of n terminals in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. The problem has applications in the design of wireless communication networks. We give a ratio-1.866 approximation algorithm for the problem.

Optimal and approximate bottleneck Steiner trees

Given a set of terminals in the plane, a bottleneck Steiner tree is a tree interconnecting the terminals, in which the length of the longest edge is minimized. The bottleneck Steiner tree problem, or special cases thereof, has applications in facility location and electronic physical design automation. In this paper, we first consider algorithms for computing optimal bottleneck Steiner trees. For a given topology, we give a direct, geometric algorithm that computes an optimal rectilinear bottleneck Steiner tree in O( n 2) time, which improves on the time complexity of previous algorithms. We also give a linear-time algorithm that, given the output from the previous algorithm, computes a rectilinear Steiner tree with minimum bottleneck length and that, among all trees with minimum bottleneck length, has minimum total length. These topology-specific algorithms provide solutions to many facility location applications, and in combination with a topology enumeration algorithm, can be used to solve the more general problems that arise in other applications. We also describe some difficulties in generalizing these results to the Euclidean problem, and give a simple approximation algorithm for the Euclidean problem. We then consider computation of approximate bottleneck Steiner trees. Specifically, we derive the exact value of the bottleneck Steiner ratio in any distance metric. The bottleneck Steiner ratio is the maximum ratio of the length of the longest edge in a minimum spanning tree to the length of the longest edge in an optimal bottleneck Steiner tree. Thus, the bottleneck Steiner ratio indicates the quality of a minimum spanning tree as an approximation of an optimal bottleneck Steiner tree.

Bottleneck Steiner trees in the plane

A Steiner tree with maximum-weight edge minimized is called a bottleneck Steiner tree (BST). The authors propose a Theta ( mod rho mod log mod rho mod ) time algorithm for constructing a BST on a point set rho , with points labeled as Steiner or demand; a lower bound, in the linear decision tree model, is also established. It is shown that if it is desired to minimize further the number of used Steiner points, then the problem becomes NP-complete. It is shown that when locations of Steiner points are not fixed the problem remains NP-complete; however, if the topology of the final tree is given, then the problem can be solved in Theta ( mod rho mod log mod rho mod ) time. The BST problem can be used, for example, in VLSI layout, communication network design, and (facility) location problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>