Steiner Problem Research Articles

Degree-Constrained Steiner Problem in Graphs with Capacity Constraints

The degree-constrained Steiner problem in graphs is well known in the literature. In an undirected graph, positive integer degree bounds are associated with nodes and positive costs with the edges. The goal is to find the minimum cost tree spanning a given node set while respecting the degree bounds. As it is known, finding a tree satisfying the constraints is not always possible. The problem differs when the nodes can participate multiple times in the coverage and the constraints represent a limited degree (a capacity) for each occurrence of the nodes. The optimum corresponds to a graph-related structure, i.e., to a hierarchy. Finding the solution to this particular Steiner problem is NP-hard. We investigate the conditions of its existence and its exact computation. The gain of the hierarchies is demonstrated by solving ILPs to compute hierarchies and trees. The advantages of the spanning hierarchies are conclusive: (1) spanning hierarchies can be found in some cases where spanning trees matching the degree constraints do not exist; (2) the cost of the hierarchy can be lower even if the Steiner tree satisfying the constraints exists.

Steiner 3-Wiener Index of Zigzag Polyhex Nanotubes.

Let G be a connected graph and S be a k element subset of the vertex set V(G) of G. Steiner distance is a natural generalization of the usual graph distance. The Steiner-k distance dG(S) between the vertices of S is the minimum size among all connected subgraphs whose vertex set contains S. The generalized indices based on Steiner distances have several applications in the real world. It is a well-known fact that "Steiner Problem" is NP-complete and hence any parameter based on Steiner distance is also an NP problem. The objective of this work is to determine the Steiner 3-Wiener index for an important chemical structure called the zigzag polyhex nanotube. In this paper, we present an algorithm for computing the Steiner 3-Wiener index (SW3) for zigzag polyhex nanotubes. The developed algorithm addresses the complexities associated with exponential increments in the number of vertices as the nanotube's circumference or length expands. The obtained SW3 values for zigzag polyhex nanotubes can be used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) analyses. We have presented an algorithm and listed the numerical values of SW3 for various parameters of circumference and length of a zigzag polyhex nanotube to facilitate their utilization in QSAR/ QSPR analyses. We have obtained the time complexity for the algorithm, which shows that the SW3 values are computationally intensive. To explain this complex nature, we have used multiple linear regression to fit log SW3 values corresponding to log p and log q, the radius and length of a nanotube. The algorithm addresses the complexities associated with the exponential increase in vertices as the nanotube's circumference or length expands. Furthermore, we provide SW3 values for various parameter combinations up to circumference or length 17, and a general relation to determine the value of SW3, facilitating its utilization in real-world applications. These values serve as crucial descriptors for understanding the structural nuances of zigzag polyhex nanotubes, that find applications in material science, drug design, drug discovery, etc.

Solution of the Steiner problem in the search for optimal network structure using population optimization methods

Subject of research: This paper considers an example of solving the Steiner problem for the design of utilities, in particular, power lines. Purpose of research: The study is electrical networks. The study is to assess the suitability of using genetic algorithms to find the network connection scheme with the lowest total active power losses. Methods and objects of research: Heuristic methods of population optimization are applied as a research method. Main results of research: As a result of the work, the suitability of applying genetic algorithms to solve the Steiner problem on the example of an electrical network to minimize total power losses has been evaluated.

New strategy for anti-loop formulations

This paper presents a strategy based on binary labelling of nodes for the creation of anti-loop formulations from existing strategies. This strategy prevents by default the formation of odd cycles, therefore it can have important role in iterative procedures based on generating subtour elimination constraints. It can also be used to modify the classic strategies used in problems associated to graphs. In this paper we focus on this last application. The behavior of this strategy is analyzed with two problems associated with graphs, the Asymmetric Traveling Salesman Problem (ATSP) and the Steiner Problem, where two configurations that modify the Miller-Tucking-Zemlig proposal to avoid cycles are compared. The experimental analysis shows that this strategy keep a good convergence, highlighting its use for the Steiner problem.

On Uniqueness in Steiner Problem

Abstract We prove that the set of $n$-point configurations for which the solution to the planar Steiner problem is not unique has the Hausdorff dimension at most $2n-1$ (as a subset of $\mathbb{R}^{2n}$). Moreover, we show that the Hausdorff dimension of the set of $n$-point configurations for which at least two locally minimal trees have the same length is also at most $2n-1$. The methods we use essentially rely upon the theory of subanalytic sets developed in [ 1]. Motivated by this approach, we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replaced by an arbitrary analytic Riemannian manifold $M$. In this setup, we argue that the set of configurations possessing two locally-minimal trees of the same length either has dimension equal to $n \dim M - 1$ or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to the above-mentioned results, we study the set of $n$-point configurations for which there is a unique solution to the Steiner problem in $\mathbb{R}^{d}$. We show that this set is path-connected.

Bussey systems and Steiner's tactical problem

In 1853, Steiner posed a number of combinatorial (tactical) problems, which eventually led to a large body of research on Steiner systems. However, solutions to Steiner's questions coincide with Steiner systems only for strengths two and three. For larger strengths, essentially only one class of solutions to Steiner's tactical problems is known, found by Bussey more than a century ago. In this paper, the relationships among Steiner systems, perfect binary one-error-correcting codes, and solutions to Steiner's tactical problem (Bussey systems) are discussed. For the latter, computational results are provided for at most 15 points.

The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems

Given a directed graph G and a list ( s 1 , t 1 ), …, ( s d , t d ) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed s i → t i path for every 1≤ i ≤ d . The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t 1 , …, t d ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every t i to every other t j ) is known to be W[1]-hard parameterized by the number of terminals. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if ℋ is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list ( s 1 , t 1 ), …, ( s d , t d ) of demands form a directed graph that is a member of ℋ. Our main result is a complete characterization of the classes ℋ resulting in fixed-parameter tractable special cases: we show that if every pattern in ℋ has the combinatorial property of being “transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges,” then the problem is FPT, and it is W[1]-hard for every recursively enumerable ℋ not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], q -Root Steiner Tree is FPT for constant q [Suchý, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [Guo et al., SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before.

The Fermat–Steiner Problem in the Space of Compact Subsets of the Euclidean Plane

The Fermat–Steiner problem is the problem of finding all points of a metric space Y such that the sum of the distances from them to points of a certain fixed finite subset A of the space Y is minimal. In this paper, we examine the Fermat–Steiner problem in the case where Y is the space of compact subsets of the Euclidean plane endowed with the Hausdorff metric, and points of A are finite pairwise disjoint compact sets.

A Self-Similar Infinite Binary Tree Is a Solution to the Steiner Problem

We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that S∪A is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties).

Pole-Aware Analog Layout Synthesis Considering Monotonic Current Flows and Wire Crossings

This article presents a new paradigm for analog placement, which further incorporates poles in addition to the considerations of symmetry island and monotonic current flow while minimizing wire crossings. The nodes along the signal paths in an analog circuit contribute to the poles, and the parasitics on these dominant poles can significantly limit the circuit performance. Although the monotonic placement methods introduced in the previous works can generate simpler routing topologies, the unawareness of poles, especially the dominant and the first non-dominant poles and wire crossings among critical nets, may increase wire load and performance degradation. This article proposes a pole-aware analog layout synthesis methodology to minimize the total wire load and wire crossings while satisfying different symmetry and topological routing constraints. Using a strong-connectivity approach to the group Steiner problem, the presented method for automatic selection of port locations can help reduce total wirelength, increase routing flexibility, and minimize total wire crossings. Experimental results show that the proposed approach results in better solution quality in circuit performance compared with other recent works.

Reliability Analysis of Survivable Networks under the Hostile Model

This article studies the Generalized Steiner Problem with Node-Connectivity Constraints and Hostile Reliability and introduces a metaheuristic resolution approach based on Greedy Randomized Adaptive Search Procedure and Variable Neighborhood Descent. Under the hostile model, nodes and links are subject to probabilistic failures. The research focuses on studying the relationship between the optimization and the reliability evaluation in a symmetric network design problem. Relevant research questions are addressed, linking the number of feasible networks for the full probabilistic model, the sensitivity with respect to elementary probabilities of operation for both edges and nodes, and the sensitivity of the model with respect to the symmetric connectivity constraints defined for terminal nodes. The main result indicates that, for the hostile model, it is better at improving the elementary probabilities of operation of links than improving the elementary probabilities of Steiner nodes, to meet a required reliability threshold for the designed network.

Heat flow for harmonic maps from graphs into Riemannian manifolds

We introduce the notion of harmonic map from a graph into a Riemannian manifold via a discrete version of the energy density. Existence and basic properties are established. Global existence and convergence of the associated heat flow are proved without any assumption on the curvature of the target manifold. We discuss a variant of the Steiner problem which replaces length by elastic energy.

The minimum degree Group Steiner problem

The DB-GST problem is given an undirected graph G(V,E), and a collection of groups S={Si}i=1q,Si⊆V, find a tree that contains at least one vertex from every group Si, so that the maximum degree is minimal. This problem was motivated by On-Line algorithms Hajiaghayi (2016), and has applications in VLSI design and fast Broadcasting. In the WDB-GST problem, every vertex v has individual degree bound dv, and every e∈E has a cost c(e)>0. The goal is, to find a tree that contains at least one terminal from every group, so that for every v, degT(v)≤dv, and among such trees, find the one with minimum cost. We give the first approximation for this problem, an (O(log2n),O(log2n)) bicriteria approximation ratio the WDB-GST problem on trees inputs. This implies an O(log2n) approximation for DB-GST on tree inputs. The previously best known ratio for the WDB-GST problem on trees was a bicriterion (O(log2n),O(log3n)) (the approximation for the degrees is O(log3n)) ratio which is folklore. Getting O(log2n) approximation requires careful case analysis and was not known.Our result for WDB-GST generalizes the classic result of Garg et al. (2016) that approximated the cost within O(log2n), but did not approximate the degree.Our main result is an O(log3n) approximation for BD-GST on Bounded Treewidth graphs.The DB-Steiner k-tree problem is given an undirected graph G(V,E), a collection of terminals S⊆V, and a number k, find a tree T(V′,E′) that contains at least k terminals, of minimum maximum degree. We prove that if the DB-GST problem admits a ρ ratio approximation, then the DB-Steiner k-tree problem, admits an O(log2k⋅ρ) expected approximation. We also show that if there are k groups, there exists an algorithm that is able to coverk/4 of the groups with minimum maximal degree, then there is a deterministic O(logn⋅ρ) approximation for DB-Steiner k-tree problem. Using the work of Guo et al. (2020) we derive an O(log3n) approximation for DB-Steiner k-tree problem on general graphs, that runs in quasi-polynomial time.

A Ginzburg–Landau model with topologically induced free discontinuities

We study a variational model which combines features of the Ginzburg-Landau model in 2D and of the Mumford-Shah functional. As in the classical Ginzburg-Landau theory, a prescribed number of point vortices appear in the small energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree 1/m with m 2 prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topological constraint. As in the Ginzburg-Landau model, the energy is parameterized by a small length scale e > 0. We perform a complete Γ-convergence analysis of the model as e ↓ 0 in the small energy regime. We then study the structure of minimizers of the limit problem. In particular, we show that the line discontinuities of a minimizer solve a variant of the Steiner problem. We finally prove that for small e > 0, the minimizers of the original problem have the same structure away from the limiting vortices.

An evolutionary design of weighted minimum networks for four points in the three-dimensional Euclidean space

Abstract We find the equations of the two interior nodes (weighted Fermat–Torricelli points) with respect to the weighted Steiner problem for four points determining a tetrahedron in R 3 \mathbb{R}^{3} . Furthermore, by applying the solution with respect to the weighted Steiner problem for a boundary tetrahedron, we calculate the twist angle between the two weighted Steiner planes formed by one edge and the line defined by the two weighted Fermat–Torricelli points and a non-neighboring edge and the line defined by the two weighted Fermat–Torricelli points. By applying the plasticity principle of quadrilaterals starting from a weighted Fermat–Torricelli tree for a boundary triangle (monad) in the sense of Leibniz, established in [A. N. Zachos, A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature, Acta Appl. Math. 129 (2014), 81–134], we develop an evolutionary scheme of a weighted network for a boundary tetrahedron in R 3 \mathbb{R}^{3} . By introducing the inverse weighted Steiner network with two interior nodes built by two different quantities of the subconscious (remaining weights) for boundary tetrahedra, we describe the evolution of a weighted network with two nodes that have inherited a subconscious. The cancellation of the dynamic plasticity of these weighted networks may be applied to the creation of evolutionary scenarios, in order to prevent the development of a quadrilateral or tetrahedral virus (a monad that has got a subconscious) and the cancerogenesis of quadrilateral cells. We continue by giving the plasticity equations for a generalized weighted minimum network with two nodes that have got a subconscious whose vertices are replaced by Euclidean spheres. This evolutionary approach may be applied to the determination of the bond strengths of molecular structures between atoms in the sense of Pauling. We obtain the analytical solutions of the weighted Fermat–Torricelli problem for the case of pairs of equal weights or one pair of equal weights. We consider as a DNA-like chain a sequence of tetrahedra whose vertices possess some symmetrical weights. By calculating the twist angles of each sequence and by applying the weighted Fermat–Torricelli tree structures with symmetrical weights or weighted Steiner tree structures, we may approximate the curve axis of a DNA-like tree chain. Finally, we construct a minimum tree, which is not a minimal Steiner tree for some boundary symmetric tetrahedra in R 3 \mathbb{R}^{3} , which has two interior nodes with equal weights having the property that the common perpendicular of some two opposite edges passes through their midpoints. We prove that the length of this minimum tree may have length less than the length of the full Steiner tree for the same boundary symmetric tetrahedra, under some angular conditions.

Optimal position of compact sets and the Steiner problem in spaces with Euclidean Gromov-Hausdorff metric

We study the geometry of the metric space of compact subsets of considered up to an orientation-preserving motion. We show that, in the optimal position of a pair of compact sets (for which the Hausdorff distance between the sets cannot be decreased), one of which is a singleton, this point is at the Chebyshev centre of the other. For orientedly similar compacta we evaluate the Euclidean Gromov-Hausdorff distance between them and prove that, in the optimal position, the Chebyshev centres of these compacta coincide. We show that every three-point metric space can be embedded isometrically in the space of compacta under consideration. We prove that, for a pair of optimally positioned compacta all compacta that lie in between in the sense of the Hausdorff metric also lie in between in the sense of the Euclidean Gromov-Hausdorff metric. For an arbitrary -point boundary formed by compact sets of a set that are neighbourhoods of segments, the Steiner point realizes the minimal filling and also belongs to the set . Bibliography: 14 titles.

A convex approach to the Gilbert–Steiner problem

We describe a convex relaxation for the Gilbert–Steiner problem both in Rd and on manifolds, extending the framework proposed in [10], and we discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces.

Variational approximation of functionals defined on 1-dimensional connected sets in ℝ n

Abstract In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in ℝ n {\mathbb{R}^{n}} . Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, SIAM J. Math. Anal. 50 2018, 6, 6307–6332], we provide a variational approximation through Ginzburg–Landau type energies proving a Γ-convergence result for n ≥ 3 {n\geq 3} .

An Efficient Rectilinear and Octilinear Steiner Minimal Tree Algorithm for Multidimensional Environments

The rectilinear/octilinear Steiner problem is the problem of connecting a set of terminals $Z$ using orthogonal and diagonal edges with minimum length. This problem has many applications, such as the EDA, VLSI circuit design, fault-tolerant routing in mesh-based broadcast, and Printed Circuit Board (PCB). This paper proposes an obstacle-avoiding 4/8/10/26-directional heuristic algorithm for this problem based on the Areibi’s concept, Higher Geometry Maze Routing, and Sollin’s minimal spanning tree algorithm. The major contributions of this paper are (1) our work is the first report for the octilinear SMTs in the multidimensional environments, (2) we provide an optimal point-to-point routing without any refinement, and (3) the proposed algorithm has higher adaptability to deal with any irregular environment, and can be extended to the $\lambda $ -geometry without any extra work, where $\lambda =2$ , 4, 8 and $\infty $ corresponding to rectilinear, 45°, 22.5° and Euclidean geometries respectively.

Glimpse of Steiner distance problem

In this paper, we review in detail the Steiner Distance problem and literature related to Steiner Distance problem in Graph and trees. The relation between the diameter and radius of Steiner problem in the graph and trees will be discuss which were studied by various researchers and the structural properties related to the Steiner distance problem.