On better heuristics for Steiner minimum trees

Finding a shortest network interconnecting a given set of points in a metric space is called the Steiner minimum tree problem. The Steiner ratio is the largest lower bound for the ratio between lengths of a Steiner minimum tree and a minimum spanning tree for the same set of points. In this paper, we show that in a metric space, if the Steiner ratio is less than one and finding a Steiner minimum tree for a set of size bounded by a fixed number can be performed in polynomial time, then there exists a polynomialtime heuristic for the Steiner minimum tree problem with performance ratio bigger than the Steiner ratio. It follows that in the Euclidean plane, there exists a polynomial-time heuristic for Steiner minimum trees with performance ratio bigger than . This solves a long-standing open problem.Part of this work was done while this author visited the Department of Computer Science, Princeton University, supported in part by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648, supported in part by NSF grant No. CCR-8920505, and also supported in part by the National Natural Science Foundation of China.     

The role of Steiner hulls in the solution to Steiner tree problems

ASteiner tree problem on the plane is that of finding a minimum lengthSteiner tree connecting a given setK ofterminals and lying within a given regionR of the Euclidean plane; it includes as special cases the Euclidean Steiner minimal tree problem (ESMT), the rectilinear Steiner tree problem (RST), and the Steiner tree problem on graphs (STG). ASteiner hull forK inR generically refers to any subregion ofR known to contain a Steiner tree. This paper gives a survey of the role of Steiner hulls in the Steiner tree problem. The significance of Steiner hulls in the efficient solution of Steiner tree problems is outlined, and then a compendium is given of the known Steiner hull constructions for ESMT, RST, and STG problems.     

Approximate Euclidean Steiner Trees

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from \(120^{\circ }\). This notion arises in numerical approximations of minimum Steiner trees. We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It has been conjectured that this relative error is at most linear in the maximum error at the angles, independent of the number of terminals. We verify this conjecture for the two-dimensional case as long as the maximum angle error is sufficiently small in terms of the number of terminals. In the two-dimensional case we derive a lower bound for the relative error in length. This bound is linear in terms of the maximum angle error when the angle error is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error in length for larger values of the maximum angle error and calculate exact values in the plane for three and four terminals.     

Approximations for two variants of the Steiner tree problem in the Euclidean plane {\mathbb{R}^2}

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 c ₁, 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, k ₁ 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 c ₂, 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, k ₂ is the number of Steiner points in T and k ₃ 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.     

Approximations for node-weighted Steiner tree in unit disk graphs

Given a node-weighted connected graph and a subset of terminals, the problem node-weighted Steiner tree (NWST) seeks a lightest tree connecting a given set of terminals in a node-weighted graph. While NWST in general graphs are as hard as Set Cover, NWST restricted to unit-disk graphs (UDGs) admits constant-approximations. Recently, Zou et al. (Lecture notes in computer science, vol 5165, COCOA, 2008, pp 278–285) showed that any μ-approximation algorithm for the classical edge-weighted Steiner tree problem can be used to produce 2.5 μ-approximation algorithm for NWST restricted to UDGs. With the best known approximation bound 1.55 for the classical Steiner tree problem, they obtained an approximation bound 3.875 for NWST restricted to UDGs. In this paper, we present three approximation algorithms for NWST restricted to UDGs, the k-Restricted Relative Greedy Algorithm whose approximation bound converges to 1 + ln 5 ≈ 2.61 as k → ∞, the 3-Restricted Greedy Algorithm with approximation bound 4\frac13{4\frac{1}{3}} , and the k-Restricted Variable Metric Algorithm whose approximation bound converges to 3.9334 as k → ∞.     

Class Steiner trees and VLSI-design

We consider a generalized version of the Steiner problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with required classes of vertices and Steiner vertices, find a shortest connected subgraph containing at least one vertex of each required class. We show that this problem is NP-hard, even if there are no Steiner vertices and the graph is a tree. Moreover, the same complexity result holds if the input class Steiner graph additionally is embedded in a unit grid, if each vertex has degree at most three, and each class consists of no more than three vertices. For similar restricted versions, we prove MAX SNP-hardness and we show that there exists no polynomial-time approximation algorithm with a constant bound on the relative error, unless P = NP. We propose two efficient heuristics computing different approximate solutions in time O(¦E¦+¦V¦log¦V¦) and in time O(c(¦E¦+¦V¦log¦V¦)), respectively, where E is the set of edges in the given graph, V is the set of vertices, and c is the number of classes. We present some promising implementation results. kw]Steiner Tree; Heuristic; Approximation complexity; MAX-SNP-hardness     

Two Heuristics for the Euclidean Steiner Tree Problem

The Euclidean Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, allowing the addition of auxiliary points to the set (Steiner points). The problem is NP-hard, so polynomial-time heuristics are desired. We present two such heuristics, both of which utilize an efficient method for computing a locally optimal tree with a given topology. The first systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by finding the Steiner tree for three of the fixed points. Then, at each iteration, it introduces a new fixed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms. This revised version was published online in July 2006 with corrections to the Cover Date.     

Polynomial Time Approximation Scheme for Symmetric Rectilinear Steiner Arborescence Problem

The Symmetric Rectilinear Steiner Arborescence (SRStA) problem is defined as follows: given a set of terminals in the positive quadrant of the plane, connect them using horizontal and vertical lines such that each terminal can be reached from the origin via a y-monotone path and the total length of all the line segments is the minimum possible. Finding an SRStA has applications in VLSI design, in data structures used in some optimization algorithms and in dynamic server problems. In this paper, we provide a polynomial time approximation scheme for the SRStA problem, improving the previous best approximation ratio of 3 for this problem.     

三维欧氏Steiner最小树的Delaunay四面体网格混合智能算法

Steiner最小树问题是组合优化中经典的NP难题,在许多实际问题中有着广泛的应用,而三维欧氏Steiner最小树问题是对二维欧氏Steiner最小树问题的推广。由于三维欧氏Steiner树问题的求解非常困难,至今为止的相关成果较为少见。本文针对该问题,利用Delaunay四面体网格剖分技术,提出了一种混合型智能求解方法,不仅可以尽量避免拓扑结构陷入局部最优,且对较大规模的问题求解亦有良好的效果。算法在Matlab环境下编程实现,经实例测试,获得了满意的效果。     

On component-size bounded Steiner trees

A Steiner tree is a tree interconnecting a given set of points in a metric space such that all leaves are given points. A (full) component of a Steiner tree is a subtree which results from splitting the Steiner tree at some given points. A k-size Steiner tree is a Steiner tree in which every component has at most k given points. The k-Steiner ratio is the largest lower bound for the ratio between lengths of a minimum Steiner tree and a minimum k-size Steiner tree for the same set of points. In this paper, we determine the 3-Steiner ratio in weighted graphs.     

Steiner Ratio for Manifolds

The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed.     

Steiner trees for fixed orientation metrics

We consider the problem of constructing Steiner minimum trees for a metric defined by a polygonal unit circle (corresponding to σ ≥ 2 weighted legal orientations in the plane). A linear-time algorithm to enumerate all angle configurations for degree three Steiner points is given. We provide a simple proof that the angle configuration for a Steiner point extends to all Steiner points in a full Steiner minimum tree, such that at most six orientations suffice for edges in a full Steiner minimum tree. We show that the concept of canonical forms originally introduced for the uniform orientation metric generalises to the fixed orientation metric. Finally, we give an O(σ n) time algorithm to compute a Steiner minimum tree for a given full Steiner topology with n terminal leaves.     

A greedy approximation algorithm for the group Steiner problem

In the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices . Each subset g_i is called a group and the vertices in ?_ig_i are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265-285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73-91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59-63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε>0, our algorithm gives an approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(max_i|g_i|)·logm) provided by the LP based approaches.     

Steiner Numbers in Graphs

Abstract

The Steiner distance d(S) of a set S of vertices in a connected graph G is the minimum size of a connected subgraph of G that contains S. The Steiner number s(G) of a connected graph G of order p is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = p—1. A smallest set S of vertices of a connected graph G of order p for which d(S) = p—1 is called a Steiner spanning set of G. It is shown that every connected graph has a unique Steiner spanning set. If G is a connected graph of order p and k is an integer with 0 ≤ k ≤ p—1, then the kth Steiner number sk(G) of G is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = k. The sequence s_o(G),s₁ (G),…,8_p-1(G) is called the Steiner sequence of G. Steiner sequences for trees are characterized.     

Rotationally optimal spanning and Steiner trees in uniform orientation metrics

We consider the problem of finding a minimum spanning and Steiner tree for a set of n points in the plane where the orientations of edge segments are restricted to λ uniformly distributed orientations, λ=2,3,4,… , and where the coordinate system can be rotated around the origin by an arbitrary angle. The most important cases with applications in VLSI design arise when λ=2 or λ=4. In the former, so-called rectilinear case, the edge segments have to be parallel to one of the coordinate axes, and in the latter, so-called octilinear case, the edge segments have to be parallel to one of the coordinate axes or to one of the lines making 45° with the coordinate axes (so-called diagonals). As the coordinate system is rotated—while the points remain stationary—the length and indeed the topology of the minimum spanning or Steiner tree changes. We suggest a straightforward polynomial-time algorithm to solve the rotational minimum spanning tree problem. We also give a simple algorithm to solve the rectilinear Steiner tree problem in the rotational setting, and a finite time algorithm for the general Steiner tree problem with λ uniform orientations. Finally, we provide some computational results indicating the average savings for different values of n and λ both for spanning and Steiner trees.     

Faster Approximation Algorithms for the Rectilinear Steiner Tree Problem

The classical Steiner tree problem requires a shortest tree spanning a given vertex subset within a graph G=(V,E). An important variant is the Steiner tree problem in rectilinear metric. Only recently two algorithms were found which achieve better approximations than the ``traditional' one with a factor of 3/2. These algorithms with an approximation ratio of 11/8 are quite slow and run in time and . A new simple implementation reduces the time to . As our main result we present efficient parametrized algorithms which reach a performance ratio of 11/8 + ɛ for any ɛ > 0 in time , and a ratio of in time . Received December 2, 1993, and in revised form July 24, 1996.     

A simple proof of Hwang's theorem for rectilinear Steiner minimal trees

We present a simple, direct proof of Hwang's characterization of rectilinear Steiner minimal trees [3]: LetS be a set of at least five terminals in the plane. If no rectilinear Steiner minimal tree forS has a terminal of degree two or more, there is a tree in which at most one of the Steiner points does not lie on a straight linel, and the tree edges incident to the Steiner points onl appear on alternate sides. This theorem has been found useful in proving other results for rectilinear Steiner minimal trees.     

Minimum Networks for Four Points in Space

The minimum network problem (Steiner tree problem) in space is much harder than the one in the Euclidean plane. The Steiner tree problem for four points in the plane has been well studied. In contrast, very few results are known on this simple Steiner problem in 3D-space. In the first part of this paper we analyze the difficulties of the Steiner problem in space. From this analysis we introduce a new concept — Simpson intersections, and derive a system of iteration formulae for computing Simpson intersections. Using Simpson intersections the Steiner points can be determined by solving quadratic equations. As well this new computational method makes it easy to check the impossibility of computing Steiner trees on 4-point sets by radicals. At the end of the first part we consider some special cases (planar and symmetric 3D-cases) that can be solved by radicals. The Steiner ratio problem is to find the minimum ratio of the length of a Steiner minimal tree to the length of a minimal spanning tree. This ratio problem in the Euclidean plane was solved by D. Z. Du and F. K. Hwang in 1990, but the problem in 3D-space is still open. In 1995 W.D. Smith and J.M. Smith conjectured that the Steiner ratio for 4-point sets in 3D-space is achieved by regular tetrahedra. In the second part of this paper, using the variational method, we give a proof of this conjecture.     

Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

We give a new lower bound on the length of the minimal Steiner tree with a given topology joining given terminals in Euclidean space, in terms of toroidal images. The lower bound is equal to the length when the topology is full. We use the lower bound to prove bounds on the “error” e in the length of an approximate Steiner tree, in terms of the maximum deviation d of an interior angle of the tree from 120°. Such bounds are useful for validating algorithms computing minimal Steiner trees. In addition we give a number of examples illustrating features of the relationship between e and d, and make a conjecture which, if true, would somewhat strengthen our bounds on the error. J. H. Rubinstein, J. Weng: Research supported by the Australian Research Council N. Wormald: Research supported by the Australian Research Council and the Canada Research Chairs Program. Research partly carried out while the author was in the Department of Mathematics and Statistics, University of Melbourne     

Degree-bounded minimum spanning trees

Given n points in the Euclidean plane, the degree-δ minimum spanning tree (MST) problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most δ. The problem is NP-hard for 2≤δ≤3, while the NP-hardness of the problem is open for δ=4. The problem is polynomial-time solvable when δ=5. By presenting an improved approximation analysis for Chan’s degree-4 MST algorithm [T. Chan, Euclidean bounded-degree spanning tree ratios, Discrete & Computational Geometry 32 (2004) 177-194], we show that, for any arbitrary collection of points in the Euclidean plane, there always exists a degree-4 spanning tree of weight at most times the weight of an MST.