Steiner Problem Research Articles

The general Steiner problem in rectangular crisscross space

The crisscross space is rectangularly structured. Start from a Cartesian coordinate system with distance D of points defined as D( P 1, P 2)=| x 2− x 1|+| y 2− y 1|, where P 1=( x 1, y 1) and P 2( x 2, y 2) are points in the real plane. The general Steiner problem is to find the minimum point P of Φ(P)= Σ i=1 n c iD(P i,p) , where p i , i=1,…, n, are given points and c i is the weight of P i . The paper concludes that (1) the 2-dimensional (as well as the n-dimensional) problem can be reduced to a pair of 1-dimensional problems: (2) the solution of the 1-dimensional problem is determined by the weights only, viz. if there is an integer k (1⩽ k⩽ n−1) such that Σ i=1 k c i= Σ i=k+1 n c i all points on the interval [ x k , x k+1 ] are minimum points. In degenerate cases, i.e. when Σ i=1 k c i= Σ i=k n c i or when neither of the above kinds of k exist, the solution reduces to a single point. In the 2-dimensional case the solution space is a rectangular region with [ x k , x k+1 ] and [ y j , y j+1 ] as sides. In a degenerate case the solution space reduces to a segment or a point.

A test problem generator for the Steiner problem in graphs

In this paper we present a new binary-programming formulation for the Steiner problem in graphs (SPG), which is well known to be NP-hard. We use this formulation to generate test problems with known optimal solutions. The technique uses the KKT optimality conditions on the corresponding quadratically constrained optimization problem.

Worst-case performance of some heuristics for Steiner's problem in directed graphs

Steiner's problem in directed graphs is to find a minimum-cost subgraph of a given directed graph such that a directed path from a root node to every vertex of a set of specified vertices exists. According to the NP-hardness of the problem one is interested in developing efficient heuristic algorithms to find good approximate solutions. In this paper we investigate the worst-case performance of some heuristics and study their comparability by means of some examples.

Directed Steiner problems with connectivity constraints

We present a generalization of the Steiner problem in a directed graph. Given nonnegative weights on the arcs, the problem is to find a minimum weight subset F of the arc set such that the subgraph induced by F contains a given number of arc-disjoint directed paths from a certain root node to each given terminal node. Some applications of the problem are discussed and properties of associated polyhedra are studied. Results from a cutting plane algorithm are reported.

The steiner problem in distributed computing systems

A distributed algorithm is presented for constructing a nearly optimal Steiner tree in an asynchronous network represented by a weighted communication graph G = ( V, E, c). The worst-case cost ratio of the obtained solution to any given minimum-cost Steiner tree T min is 2( 1−1 l) , where l is the number of leaves of T min . The message complexity of the algorithm is O(|E| + |V|∗(|V| + log|V|)) and the time complexity is O| V|*| V|), where S is the subset of nodes of G to be connected.

An 11/6-approximation algorithm for the network steiner problem

An instance of the Network Steiner Problem consists of an undirected graph with edge lengths and a subset of vertices; the goal is to find a minimum cost Steiner tree of the given subset (i.e., minimum cost subset of edges which spans it). An 11/6-approximation algorithm for this problem is given. The approximate Steiner tree can be computed in the time0(¦V¦ ¦E¦ + ¦S¦4), whereV is the vertex set,E is the edge set of the graph, andS is the given subset of vertices.

THE STEINER PROBLEM IN THE PLANE OR IN PLANE MINIMAL NETS

The famous Steiner problem in the Euclidean plane, which is that of investigating minimal nets spanning fixed finite subsets M of points in the plane, is solved when M is extremal, i.e. when M lies on the boundary of its convex hull, and the nets are nondegenerate, i.e. have no vertices of degree 2.

An integer programming formulation of the Steiner problem in graphs

A succinct integer linear programming model for the Steiner problem in networks is presented.

The Steiner problem: a survey

AbstractFor the past decade the Steiner minimal tree problem has attracted the attention of researchers in discrete optimization. A brief survey of the main results concerning the properties and algorithms of the Steiner problem in the Euclidean plane, the Steiner problem in the plane with rectilinear metric and the Steiner problem in networks is done in this paper. The main attention is paid to the recent results concerning the last problem.

Reducing the Steiner Problem in a Normed Space

Consider a set P of points in a normed space whose unit sphere is a d-dimensional symmetric polytope with 2d extreme points. This paper proves that there always exists a Steiner minimum tree whose Steiner points are located only at points whose coordinates appear in points of P. This generalizes a recent result of Snyder on d-dimensional rectilinear space, which itself extends Hanan’s well-known and much quoted result on the rectilinear plane. Furthermore, the proof in this paper is much simpler than Snyder’s proof, even considerably shorter than Hanan’s proof. A consequence of this result is that the Steiner problem for P in such a space is reduced to a Steiner problem on graphs and is solvable by any existing Steiner graph algorithms. The paper also conjectures that such a reduction is impossible if the polytope has more than $2d$ extreme points and provides partial support for the conjecture.

Steiner's problem in graphs: heuristic methods

Real world problems arising in the layout of connection structures in networks as e.g. in VLSI design may often be decomposed into a number of well-known combinatorial optimization problems. Steiner's problem in graphs is included within this context. According to its complexity one is interested in developing efficient heuristic algorithms to find good approximate solutions. Here a comparison of various heuristic methods for Steiner's problem in graphs is presented.

Degenerate Gilbert—Steiner trees

The minimal Gilbert network problem is a generalization of the Steiner minimal tree problem derived by adding flow-dependent weights to the edges. The minimal Gilbert network is referred to as a minimal Gilbert—Steiner tree if it has a Steiner topology. All trees with Steiner topologies, obtained by sequentially contracting edges connecting given points and their incident Steiner points, form a degenerate tree family. Similarly to the Steiner problem, a relatively minimal tree satisfying some angle conditions is referred to as a Gilbert–Steiner tree. This work shows that, unlike the Steiner problem, there may exist more than one Gilbert–Steiner tree in a degenerate tree family. It is proved that the maximum number of Gilbert–Steiner trees in a degenerate tree family is 2[(n−2)/2]. A necessary condition for the minimal Gilbert–Steiner tree is also given.

Heuristics for the Steiner problem in graphs

The Steiner problem in graphs (networks) is to find a minimum cost tree spanning a given subset Z of vertices in a graph G with positive edge costs. We present a new worst-case performance analysis of the contraction heuristic. Also an improved version of this heuristic is suggested and analysed. Our main contribution is a proof of certain incomparability of some known heuristics. We present examples for which the heuristics extremely outperform each other in terms of the quality of the solution.

Path-distance heuristics for the Steiner problem in undirected networks

An integrative overview of the algorithmic characteristics of three well-known polynomialtime heuristics for the undirected Steiner minimum tree problem:shortest path heuristic (SPH),distance network heuristic (DNH), andaverage distance heuristic (ADH) is given. The performance of thesesingle-pass heuristics (and some variants) is compared and contrasted with several heuristics based onrepetitive applications of the SPH. It is shown that two of these repetitive SPH variants generate solutions that in general are better than solutions obtained by any single-pass heuristic. The worst-case time complexity of the two new variants isO(pn 3) andO(p 3 n 2), while the worst-case time complexity of the SPH, DNH, and ADH is respectivelyO(pn 2),O(m + n logn), andO(n 3) wherep is the number of vertices to be spanned,n is the total number of vertices, andm is the total number of edges. However, use of few simple tests is shown to provide large reductions of problem instances (both in terms of vertices and in term of edges). As a consequence, a substantial speed-up is obtained so that the repetitive variants are also competitive with respect to running times.

Problems with generalized steiner problems

Problems with generalized steiner problems

A heuristic for Euclidean and rectilinear Steiner problems

In this paper we present a heuristic for Euclidean and rectilinear Steiner problems. This heuristic is based upon finding optimal Steiner solutions for connected subgraphs of the minimal spanning tree of the entire vertex set. Computational results are given for randomly generated problems involving up to 10000 vertices.

On the Exact Location of Steiner Points in General Dimension

The Steiner Problem is to form a minimum-length tree that contains a given set of points, where augmentation of the point set with additional (Steiner) points is permitted. The Rectilinear Steiner Problem is one in which edge weights are determined by the $L_1 $, or Manhattan, distance between points in $\mathbb{R}^d $. Let S be a set of n points $\mathbb{R}^d $, where $d \geq 2$. By generalizing a planar theorem of Hanan (SIAM J. Appl. Math., 14 (1966), pp. 255–265.) to all dimensions, a set of $O(n^d )$ points that is guaranteed to include all the Steiner points of a minimum rectilinear Steiner tree of S is constructed, complementing Hanan’s result in $d = 2$. The theorem here and its proof illuminate new combinatorial and geometrical facts about rectilinear Steiner trees; one immediate benefit is algorithms that are asymptotically faster than all known algorithms for the problem in dimensions three and greater. The theorem also yields a polynomial algorithm in all dimensions for a version of the problem known as the Rectilinear k-Steiner Problem.

The steiner problem in the hypercube

AbstractLet Q(n) be the n‐dimensional hypercube, and X, a set of points in Q(n). The Steiner problem for the hypercube is to find the smallest possible number L(n,X) of edges in any subtree of Q(n) that spans X. We obtain the following results: An exact formula for L(n,X), when |X| ≤ 5. The bound L(n,X) ≤ (nk+1) + (2 + o(1)) ([log (k)]/k)(nk) as k → ∞, when X is the set of all points in Q(n) of a given weight k + 1, provided (k2/[log (k)])1 + 1/k ≤ n. NP‐completeness of deciding L(n,X) even when every point of X has weight at most 2.

The Steiner ratio conjecture for six points

The Steiner problem is to find a shortest network (tree) S in the plane R 2 connecting a given set X of n points. Let T be a shortest tree connecting the points of X and with vertices only at these points. T is called a minimal spanning tree. Let L S and L T denote the lengths of S and T, respectively, and let ρ=L S /L T . ρ is called the Steiner ratio. In this paper one proves the Steiner ratio conjecture for six points. One uses a variational approach

A Comparison of Two Simulated Annealing Algorithms Applied to the Directed Steiner Problem on Networks

The well-known Steiner problem on networks is an NP-complete problem for which there are many deterministic algorithms and heuristics. In this paper a new approach to the directed version of this problem is made by applying the ideas of statistical mechanics through the use of the method of simulated annealing. Two different types of cooling algorithms are tailored to the Directed Steiner Problem. Computations are done on a test bed of 480 random graphs in order to study the performance of these algorithms. In addition to the quality of the final answers, the study looks at the distribution of first occurrences during the execution of the cooling schedules of answers that are within 3% of the optimum. The study suggests that these near-optimal answers appear according to an approximately exponential distribution and that they can be obtained in polynomial time. Experiments also indicate that for random graphs annealing usually gives near-optimal answers in less time than does branch and bound. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.