An improved exact algorithm for the Euclidean k-Steiner tree problem

The Steiner tree problem is a well known network optimization problem which asks for a connected minimum network (called a Steiner minimum tree) spanning a given point set N. In the original Steiner tree problem the given points lie in the Euclidean plane or space, and the problem has many variants in different applications now. Recently a new type of Steiner minimum tree, probability Steiner minimum tree, is introduced by the authors in the study of phylogenies. A Steiner tree is a probability Steiner tree if all points in the tree are probability vectors in a vector space. The points in a Steiner minimum tree (or a probability Steiner tree) that are not in the given point set are called Steiner points (or probability Steiner points respectively). In this paper we investigate the properties of Steiner points and probability Steiner points, and derive the formulae for computing Steiner points and probability Steiner points in ℓ1- and ℓ2-metric spaces. Moreover, we show by an example that the length of a probability Steiner tree on 3 points and the probability Steiner point in the tree are smooth functions with respect to p in d-space.

High throughput genomic/proteomic strategies, such as microarray studies, drug screens, and genetic screens, often produce a list of genes that are believed to be important for one or more reasons. Unfortunately it is often difficult to discern meaningful biological relationships from such lists. This study presents a new bioinformatic approach that can be used to identify regulatory subnetworks for lists of significant genes or proteins. We demonstrate the utility of this approach using an interaction network for yeast constructed from BIND, TRANSFAC, SCPD, and chromatin immunoprecipitation (ChIP)-Chip data bases and lists of genes from well known metabolic pathways or differential expression experiments. The approach accurately rediscovers known regulatory elements of the heat shock response as well as the gluconeogenesis, galactose, glycolysis, and glucose fermentation pathways in yeast. We also find evidence supporting a previous conjecture that approximately half of the enzymes in a metabolic pathway are transcriptionally co-regulated. Finally we demonstrate a previously unknown connection between GAL80 and the diauxic shift in yeast.

This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light $$k$$k-Steiner tree problem (SL$$k$$kST), we are given an undirected graph $$G=(V,E)$$G=(V,E) with terminals $$T\subseteq V$$T⊆V containing a root $$r\in T$$r?T, a cost function $$c:E\rightarrow \mathbb {R}^+$$c:E?R+, a length function $$\ell :E\rightarrow \mathbb {R}^+$$l:E?R+, a bound $$L>0$$L>0 and an integer $$k\ge 1$$k?1. The goal is to find a minimum $$c$$c-cost $$r$$r-rooted Steiner tree containing at least $$k$$k terminals whose diameter under $$\ell $$l metric is at most $$L$$L. The input to the buy-at-bulk $$k$$k-Steiner tree problem (BB$$k$$kST) is similar: graph $$G=(V,E)$$G=(V,E), terminals $$T\subseteq V$$T⊆V containing a root $$r\in T$$r?T, cost and length functions $$c,\ell :E\rightarrow \mathbb {R}^+$$c,l:E?R+, and an integer $$k\ge 1$$k?1. The goal is to find a minimum total cost $$r$$r-rooted Steiner tree $$H$$H containing at least $$k$$k terminals, where the cost of each edge $$e$$e is $$c(e)+\ell (e)\cdot f(e)$$c(e)+l(e)·f(e) where $$f(e)$$f(e) denotes the number of terminals whose path to root in $$H$$H contains edge $$e$$e. We present a bicriteria $$(O(\log ^2 n),O(\log n))$$(O(log2n),O(logn))-approximation for SL$$k$$kST: the algorithm finds a $$k$$k-Steiner tree with cost at most $$O(\log ^2 n\cdot \text{ opt }^*)$$O(log2n·opt?) where $$\text{ opt }^*$$opt? is the cost of an LP relaxation of the problem and diameter at most $$O(L\cdot \log n)$$O(L·logn). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX'06/Algorithmica'09) which had ratio $$(O(\log ^4 n), O(\log ^2 n))$$(O(log4n),O(log2n)). Using this, we obtain an $$O(\log ^3 n)$$O(log3n)-approximation for BB$$k$$kST, which improves upon the $$O(\log ^4 n)$$O(log4n)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost $$2$$2-edge-connected subgraph with at least $$k$$k vertices, which is introduced as the $$(k,2)$$(k,2)-subgraph problem in Lau et al. (2009) (STOC'07/SICOMP09). This generalizes some well-studied classical problems such as the $$k$$k-MST and the minimum cost $$2$$2-edge-connected subgraph problems. We give an $$O(\log n)$$O(logn)-approximation algorithm for this problem which improves upon the $$O(\log ^2 n)$$O(log2n)-approximation algorithm of Lau et al. (2009).

We consider Steiner’s Problem in \( \mathcal{L}_p^3 \) , which is a three-dimensional space equipped with p-norm. Steiner’s Problem is the “Problem of shortest connectivity”, that means, given a finite set N of points in the plane, search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which axe to be connected. Such points are called Steiner points. If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST) for N. Steiner’s Problem is very hard as well in combinatorial as in computational sense, but on the other hand, the determination of an MST is simple. Consequently, we are interested in the greatest lower bound for the ratio between the lengths of these both trees: $$ m(3,p): = inf\left\{ {\frac{{L(SMT for N)}} {{L(MST for N)}}:N \subseteq {\text{ }}\mathcal{L}_p^3 is a finite set} \right\} $$ , which is called the Steiner ratio (of \( \mathcal{L}_p^3 \) ,).

Given n terminals in the Euclidean plane and a positive constant l, find a Steiner tree T interconnecting all terminals with the minimum total cost of Steiner points and a specific material used to construct all edges in T such that the Euclidean length of each edge in T is no more than l. In this paper, according to the cost b of each Steiner point and the different costs of some specific materials with the different lengths, we study two variants of the Steiner tree problem in the Euclidean plane as follows: (1) If a specific material to construct all edges in such a Steiner tree has its infinite length and the cost of per unit length of such a specific material used is c1, the objective is to minimize the total cost of the Steiner points and such a specific material used to construct all edges in T, i.e., \({{\rm min} \{b \cdot k_1+ c_1 \cdot \sum_{e \in T} w(e)\}}\), where T is a Steiner tree constructed, k1 is the number of Steiner points and w(e) is the length of part cut from such a specific material to construct edge e in T, and we call this version as the minimum-cost Steiner points and edges problem (MCSPE, for short). (2) If a specific material to construct all edges in such a Steiner tree has its finite length L (l ≤ L) and the cost of per piece of such a specific material used is c2, the objective is to minimize the total cost of the Steiner points and the pieces of such a specific material used to construct all edges in T, i.e., \({{\rm min} \{b \cdot k_2+ c_2 \cdot k_3\}}\), where T is a Steiner tree constructed, k2 is the number of Steiner points in T and k3 is the number of pieces of such a specific material used, and we call this version as the minimum-cost Steiner points and pieces of specific material problem (MCSPPSM, for short). These two variants of the Steiner tree problem are NP-hard with some applications in VLSI design, WDM optical networks and wireless communications. In this paper, we first design an approximation algorithm with performance ratio 3 for the MCSPE problem, and then present two approximation algorithms with performance ratios 4 and 3.236 for the MCSPPSM problem, respectively.

Steiner Trees for Ladders

Given a set β of points in the Euclidean plane, the standard Steiner Problem asks for a network of minimum length connecting the points of β. A solution to this problem is called a Steiner Minimal Tree and may be computed exactly, using Melzak's compass-and-straightedge methods. A Steiner Tree often contains additional nodes, called Steiner points, not belonging to β. The node-weighted Steiner problem asks for the network connecting the points of β which minimizes the energy E = length + α(number of Steiner points), where α is a fixed nonnegative weight. Solutions to this problem, here called node-weighted Steiner trees, can have different geometries than those available to standard Steiner trees. This paper presents a modified Melzak procedure which computes the node-weighted Steiner tree for a given set β. © 1996 John Wiley & Sons, Inc.

The Steiner problem is to find a shortest connection of a finite subset X in a given metric space (M, d). If the space meets some local compactness and connectivity conditions, then a solution, a Steiner minimal tree for X, exists [7]. More generally, considering any graph-theoretic tree T with all nodes of degree < 3 labeled by elements of X (that is, an X-labeled tree), we may ask for a minimal length realization of T in (M, d), that is, for an embedding of the node set of T in M which extends the identity map of X and yields the smallest possible length of the image Steiner tree (relative to the labeled tree T). The embedded unlabeled nodes of T are called Steiner points. A Steiner minimal tree is then that realization which attains minimum length among the (finite) collection of X-labeled trees. Given the subset X, one is interested in restricting the search for Steiner points in the metric space (M, d). Any proper subset of M that is guaranteed to harbor at least one Steiner minimal tree for X is called a Steiner hull for X 7,12. Particular interest attaches to Steiner hulls that are finitely generated in that they are determined by a finite subset S of M containing X and all Steiner points of some Steiner minimal tree for X; this finite set S can trivially be turned into a Steiner hull by attaching one geodesic from M for each pair of its points. For the Steiner problem in rectilinear space, that is, a d-dimensional real space equipped with the metric associated with the 1-norm, finitely generated Steiner hulls have been described 7,12.KeywordsSteiner TreeSteiner PointInterior NodeSteiner Tree ProblemMedian OperationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

LetS = {A, B, C, D} consist of the four corner points of a convex quadrilateral where diagonals [A, C] and [B, D] intersect at the pointO. There are two possible full Steiner trees forS, theAB-CD tree hasA andB adjacent to one Steiner point, andC andD to another; theAD-BC tree hasA andD adjacent to one Steiner point, andB andC to another. Pollak proved that if both full Steiner trees exist, then theAB-CD (AD-BC) tree is the Steiner minimal tree if AOD>3 (<) 90°, and both are Steiner minimal trees if AOD=90°. While the theorem has been crucially used in obtaining results on Steiner minimal trees in general, its applicability is sometimes restricted because of the condition that both full Steiner trees must exist. In this paper we remove this obstacle by showing: (i) Necessary and sufficient conditions for the existence of either full Steiner tree forS. (ii) If AOD≥90°, then theAB-CD tree is the SMT even if theAD-BC tree does not exist. (iii) If AOD<90° but theAD-BC tree does not exist, then theAB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can.

Let V V denote a given set of n n points in the euclidean plane. A Steiner minimal tree for V V is the shortest network (clearly, it has to be a tree) interconnecting V V . Junctions of the network which are not in V V are called Steiner points (those in V V will be called regular points). A shortest tree interconnecting V V without using any Steiner points is called a minimal tree. Let σ ( V ) \sigma (V) and μ ( V ) \mu (V) denote the lengths of a Steiner minimal tree and a minimal tree, respectively. Define ρ \rho to be the greatest lower bound for the ratio σ ( V ) / μ ( V ) \sigma (V)/\mu (V) over all V V . We prove ρ &gt; .8 \rho &gt; .8 .

The wind farm layout optimization problem is similar to the classical mathematical problem of finding the Steiner Minimal Tree Problem (SMTP) of a weighted undirected graph. Due to the cable current-carrying capacity limitation, the cable sectional area should be carefully selected to meet the system operational requirement and this constraint should be considered during the SMTP formulation process. Hence, traditional SMTP algorithm cannot ensure a minimal cable investment layout. In this paper, a hybrid algorithm based on modified Ants Colony Optimization (ACO) and Artificial Immune Algorithm (AIA) for solving SMTP is introduced. Since the Steiner Tree Problem is NP-hard, we design an algorithm to construct high quality Steiner trees in a short time which is suitable for real time multicast routing in networks. After the breadth - first traversal of the minimal graph obtained by ACO, the terminal points are divided into different convex hull sets, and the full Steiner tree is structured from the convex hull sets partition. The Steiner points can then be vaccinated by AIA to get an optimal graph. The average optimization effect of AIA is shorter than the minimal graph obtained using ACO, and the performance of the algorithm is shown. We give an example of application in optimization for onshore wind farm Cable. The possibility of using different sectional area’s cable is also considered in this paper.

In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the $\ell_1$-metric (and Hamming metric). Chleb\'ik and Chleb\'ikov\'a [TCS'08] showed that DST is NP-hard to approximate to factor of $96/95\approx 1.01$ in the graph metric (and consequently $\ell_\infty$-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric. In this work, we prove that DST is APX-hard in every $\ell_p$-metric. We also prove that CST is APX-hard in the $\ell_{\infty}$-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in $\ell_p$-metrics.Comment: Abstract shortened. Journal version for TheoretiCS

The underlying models of many practical problems in various engineering fields are equivalent to the Steiner tree problem in graphs, which is a typical NP-hard combinatorial optimization problem. Thus, developing a fast and effective heuristic for the Steiner tree problem in graphs is of universal significance. By analyzing the advantages and disadvantages of the fast classic heuristics, we find that the shortest paths and Steiner points play important roles in solving the Steiner tree problem in graphs. Based on the analyses, we propose a Steiner point candidate-based heuristic algorithm framework (SPCF) for solving the Steiner tree problem in graphs. SPCF consists of four stages: marking [Formula: see text] points, constructing the Steiner tree, eliminating the detour paths, and [Formula: see text]-based refining stage. For each procedure of SPCF, we present several alternative strategies to make the trade-off between the effectiveness and efficiency of the algorithm. By finding the shortest path clusters between vertex sets, several methods are proposed to mark the first type of Steiner point candidates [Formula: see text]. The solution qualities of the classic heuristics are effectively improved by looking [Formula: see text] points as terminals. By constructing a Voronoi diagram, a series of methods are suggested to mark the second type of Steiner point candidates [Formula: see text]. The feasible solution quality is efficiently improved by employing the [Formula: see text] points as the insertable key-vertices in key-vertex insertion local search method. Numerical experiments show that the proposed strategies are all effective for improving the solution quality. Compared with other effective algorithms, the proposed algorithms can achieve better solution quality and speed performance.

We consider the Steiner Problem in L p 2 , which is a plane equiped with the p-norm. Steiner’s Problem is the “Problem of shortest connectivity”, that means, given a finite set N of points in the plane, search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which are to be connected. Such points are called Steiner points. If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST) for N.