Steiner Problem Research Articles

Calibrations for minimal networks in a covering space setting

In this paper, we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover, we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.

Numerical approximation of the Steiner problem in dimension $2$ and $3$

The aim of this work is to present some numerical computations of solutions of the Steiner Problem , based on the recent phase field approximations proposed in [12] and analyzed in [5, 4]. Our strategy consists in improving the regularity of the associated phase field solution by use of higher-order derivatives in the Cahn-Hilliard functional as in [6]. We justify the convergence of this slightly modified version of the functional, together with other technics that we employ to improve the numerical experiments. In particular, we are able to consider a large number of points in dimension 2. We finally present and justify an approximation method that is efficient in dimension 3, which is one of the major novelties of the paper.

On different notions of calibrations for minimal partitions and minimal networks in ℝ2

Abstract Calibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces and adapted to the case of the Steiner problem in several variants. Our goal is to compare the different notions of calibrations for the Steiner problem and for planar minimal partitions that are already present in the literature. The paper is then complemented with remarks on the convexification of the problem, on nonexistence of calibrations and on calibrations in families.

Insight into the computation of Steiner minimal trees in Euclidean space of general dimension

We present well known properties related to the topology of Steiner minimal trees and to the geometric position of Steiner points, and investigate their application in the exact algorithms that have been proposed for the Euclidean Steiner problem. We discuss the difficulty in the application of properties that were very successfully applied to solve the problem in the plane, when the dimension of the space increases, and point out that the application of some geometric conditions for Steiner points is hindered when the well known implicit enumeration scheme proposed by Smith in 1992 is considered. Finally, we mention mathematical-optimization methods as a direction to explore in the search for good formulations of inequalities that would allow the application of these restrictive geometric conditions.

Steiner Problem in the Gromov-Hausdorff Space: The Case of Finite Metric Spaces

The Steiner problem is considered in the Gromov-Hausdorff space, i.e., in the space of compact metric spaces (considered up to isometry) endowed with the Gromov-Hausdorff distance. Since this space is not boundedly compact, the problem of the existence of a shortest network in this space is open. It is shown that each finite family of finite metric spaces can be connected by a shortest network. Moreover, it turns out that in this case there exists a shortest tree all of whose vertices are finite metric spaces. An estimate for the number of points in these metric spaces is obtained. As an example, the case of three-point metric spaces is considered. It is also shown that the Gromov-Hausdorff space does not realize minimal fillings; i.e., shortest trees in this space need not be minimal fillings of their boundaries.

Optimization of the method of constructing reference plans of multimodal transport problem

The classical transport problem is in determination of the optimal plan for the transportation of goods from the points of departure to the points of delivery, taking into account the criterion of the minimum cost of such transportation. Such a problem takes into account only one type of transport, which does not fully correspond to the practical needs of modern logistics enterprises. That is why the object of this research is the classical transport problem, the formulation of which takes into account the presence of several means of cargo delivery, namely: automobile, railway and water. This type of transport problem is defined as multimodal. The implementation of the multimodal transport problem involves the use of various numerical methods and is carried out using software. In fact, the conceptual approach to its solution is a simple selection of possible results. Given the large dimension of the problem, such an approach can be extremely cumbersome, and therefore requires some improvement. During the study, the method for constructing a reference plan for such a problem was optimized based on the criterion of minimizing the number of numerical iterations, and the advantages of the proposed approach compared to those already known were substantiated. The basis of the new approach is the previously known minimal element method, which is to be used to solve the transportation problem, and an analogy with the Steiner problem was drawn. The latter, in turn, made it possible to define a new approach as the Steiner method. The research result is development of a general algorithm for the implementation of the proposed Steiner method. As an approbation of this algorithm, a model example is provided. It demonstrated the identity of the results of solving a multimodal transport problem using all the methods discussed in the article. The development of new methods for the implementation of the multimodal transport problem will make it possible to construct efficient algorithms for solving more complex problems of transport logistics. The criterion for reducing the number of numerical iterations, used at all stages of the implementation of such problems, significantly reduces the time to search for their solutions.

Convex relaxation and variational approximation of functionals defined on 1-dimensional connected sets

In this short note we announce the main results of [2] about variational problems involving 1-dimensional connected sets in the Euclidean plane, such as for example the Steiner tree problem and the irrigation (Gilbert–Steiner) problem.

Calibrations in families for minimal networks

AbstractWe define the calibrations in families for minimal Steiner networks and we explain through an example its convenience with respect to the “classical” notions of calibrations for the Steiner Problem.

Reduction techniques for the prize collecting Steiner tree problem and the maximum‐weight connected subgraph problem

AbstractThe concept of reduction has frequently distinguished itself as a pivotal ingredient of exact solving approaches for the Steiner tree problem in graphs. In this article we broaden the focus and consider reduction techniques for three Steiner problem variants that have been extensively discussed in the literature and entail various practical applications: The prize‐collecting Steiner tree problem, the rooted prize‐collecting Steiner tree problem and the maximum‐weight connected subgraph problem. By introducing and subsequently deploying numerous new reduction methods, we are able to drastically decrease the size of a large number of benchmark instances, already solving more than 90% of them to optimality. Furthermore, we demonstrate the impact of these techniques on exact solving, using the example of the state‐of‐the‐art Steiner problem solver SCIP‐Jack.

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted undirected graph G embedded on a surface of genus g and a designated face f bounded by a simple cycle of length k , uncovers a set F ⊆ E ( G ) of size polynomial in g and k that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of f . We apply this general theorem to prove that: — Given an unweighted graph G embedded on a surface of genus g and a terminal set S ⊆ V ( G ), one can in polynomial time find a set F ⊆ E ( G ) that contains an optimal Steiner tree T for S and that has size polynomial in g and | E ( T )|. — An analogous result holds for an optimal Steiner forest for a set S of terminal pairs. — Given an unweighted planar graph G and a terminal set S ⊆ V ( G ), one can in polynomial time find a set F ⊆ E ( G ) that contains an optimal (edge) multiway cut C separating S (i.e., a cutset that intersects any path with endpoints in different terminals from S ) and that has size polynomial in | C |. In the language of parameterized complexity, these results imply the first polynomial kernels for S teiner T ree and S teiner F orest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (E dge ) M ultiway C ut on planar graphs (parameterized by the size of the cutset). Additionally, we obtain a weighted variant of our main contribution: a polynomial-time algorithm that, given an undirected plane graph G with positive edge weights, a designated face f bounded by a simple cycle of weight w ( f ), and an accuracy parameter ε > 0, uncovers a set F ⊆ E ( G ) of total weight at most poly(ε -1 ) w ( f ) that, for any set of terminal pairs that lie on f , contains a Steiner forest within additive error ε w ( f ) from the optimal Steiner forest.

Regularity of Maximum Distance Minimizers

We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality max yϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline. In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.

On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers

In this article, we consider and analyse a variant of a functional originally introduced in \[9, 27] to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter $\epsilon > 0$ and resembles the (scalar) Ginzburg–Landau functional from phase transitions. In a first part, we prove existence and regularity of minimizers for this functional. Then we provide a detailed analysis of their behavior as $\epsilon \to 0$, showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem and optimal compliance are also discussed.

Convex relaxation and variational approximation of the Steiner problem: theory and numerics

Abstract We survey some recent results on convex relaxations and a variational approximation for the classical Euclidean Steiner tree problem and we see how these new perspectives can lead to effective numerical schemes for the identification of Steiner minimal trees.

The Use of the Adapted Task of Steiner Problem for the Solution of Optimization Problems of Realization of Industrial Enterprises Production

The Use of the Adapted Task of Steiner Problem for the Solution of Optimization Problems of Realization of Industrial Enterprises Production

Improved Approximation Algorithms for (Budgeted) Node-weighted Steiner Problems

Moss and Rabani study constrained node-weighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an $O(\log n)$-approximation algorithm for the node-weighted prize-collecting Steiner tree problem (PCST)---where the goal is to minimize the cost of a tree plus the penalty of vertices not covered by the tree. They use the algorithm for PCST to obtain a bicriteria $(2, O(\log n))$-approximation algorithm for the budgeted node-weighted Steiner tree problem---where the goal is to maximize the prize of a tree with a given budget for its cost. Their solution may cost up to twice the budget, but collects a factor $\Omega(\frac{1}{\log n})$ of the optimal prize. We improve these results from at least two aspects. Our first main result is a primal-dual $O(\log h)$-approximation algorithm for a more general problem, node-weighted prize-collecting Steiner forest (PCSF), where we have $h$ demands each requesting the connectivity of a pair of vertices. Our algorithm can be seen as a greedy algorithm which reduces the number of demands by choosing a structure with minimum cost-to-reduction ratio. This natural style of argument leads to a much simpler algorithm than that of Moss and Rabani for PCST. Our second main contribution is for the budgeted node-weighted Steiner tree problem, which is also an improvement to the work of Moss and Rabani. In the unrooted case, we improve upon an existing $O(\log^2 n)$-approximation by Guha et al., and present an $O(\log n)$-approximation algorithm without any budget violation. For the rooted case, where a specified vertex has to appear in the solution tree, we improve the bicriteria result of Moss and Rabani to the bicriteria approximation ratio of $(1+\epsilon, O(\log n)/\epsilon^2)$ for any positive (possibly subconstant) $\epsilon$. That is, for any permissible budget violation $1+\epsilon$, we present an algorithm achieving a trade off in the guarantee for the prize. Indeed, we show that this is almost tight for the natural linear-programming relaxation used by us as well as in the previous works.

A robust and scalable algorithm for the Steiner problem in graphs

We present an effective heuristic for the Steiner Problem in Graphs. Its main elements are a multistart algorithm coupled with aggressive combination of elite solutions, both leveraging recently-proposed fast local searches. We also propose a fast implementation of a well-known dual ascent algorithm that not only makes our heuristics more robust (by dealing with easier cases quickly), but can also be used as a building block of an exact branch-and-bound algorithm that is quite effective for some inputs. On all graph classes we consider, our heuristic is competitive with (and sometimes more effective than) any previous approach with similar running times. It is also scalable: with long runs, we could improve or match the best published results for most open instances in the literature.

A phase-field approximation of the Steiner problem in dimension two

AbstractIn this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form1+β⁢θ{1+\beta\,\theta}, where θ denotes the amount of transported mass andβ>0{\beta>0}is a fixed parameter (notice that the limit caseβ=0{\beta=0}corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({ℱε}ε>0{\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}}) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of{ℱε}{\{\mathcal{F}_{\varepsilon}\}}asε↓0{\varepsilon\downarrow 0}. Our functionals are modeled on the Ambrosio–Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

A node‐based ILP formulation for the node‐weighted dominating Steiner problem

In this article, we consider the Node‐Weighted Dominating Steiner Problem. Given a graph with node weights and a set of terminal nodes, the goal is to find a connected node‐induced subgraph of minimum weight, such that each terminal node is contained in or adjacent to some node in the chosen subgraph. The problem arises in applications in the design of telecommunication networks. Integer programming formulations for Steiner problems usually employ a variable for each edge. We introduce a formulation that only uses node variables and that models connectivity through node‐cut inequalities, which can be separated in polynomial time. We discuss necessary and sufficient conditions for the model inequalities to define facets and we introduce a class of lifted partition‐based inequalities, which can be used to strengthen the linear relaxation. Finally, we show that the polyhedron defined by these inequalities is integral if the underlying graph is a cycle where no two terminals are adjacent. In the general cycle setting, we show that we can get a complete description of the feasible solutions by lifting and projecting into a polytope with no more than twice the dimension. We also show that the well‐known indegree equalities are implied by the lifted partition inequalities. Finally, we evaluate the effectiveness of the presented partition inequalities in computational experiments. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 69(1), 33–51 2017

Fermat–Steiner problem in the metric space of compact sets endowed with Hausdorff distance

Fermat–Steiner problem consists in finding all points in a metric space Y such that the sum of distances from each of them to the points from some fixed finite subset A of Y is minimal. Such points are sometimes referred as geometric medians of A. This problem is investigated for the metric space \(Y=H(X)\) of compact subsets of a metric space X, endowed with the Hausdorff distance. For the case of a proper metric space X a description of all compacts \(K\in H(X)\) which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct a surprising example of a quite symmetric regular triangle in \(H(\mathbb {R}^2)\), such that all its shortest trees have no “natural” symmetry.

On sets minimizing their weighted length in uniformly convex separable Banach spaces

We study existence and partial regularity relative to the weighted Steiner problem in Banach spaces. We show C1 regularity almost everywhere for almost minimizing sets in uniformly rotund Banach spaces whose modulus of uniform convexity verifies a Dini growth condition.