The Euclidean k-Steiner tree problem asks for a minimum-cost network connecting n given points in the plane, allowing at most k additional nodes referred to as Steiner points. In the classical Steiner tree problem in which there is no restriction on the number of nodes, every Steiner point must be of degree 3. The k-Steiner problem differs in that Steiner points of degree 4 may be included in an optimal solution. This simple change leads to a number of complexities when attempting to create a generation algorithm for optimal k-Steiner trees, which has proven to be a powerful component of the flagship algorithm, namely GeoSteiner, for solving the classical Steiner tree problem. In the present paper we firstly extend the basic framework of GeoSteiner's generation algorithm to include degree-4 Steiner points. We then introduce a number of novel results restricting the structural and geometric properties of optimal k-Steiner trees, and then show how these properties may be used as topological pruning methods underpinning our generation algorithm. Finally, we present experimental data to show the effectiveness of our pruning methods in reducing the number of sub-optimal solution topologies.

This paper introduces an exact algorithm for the construction of a shortest curvature-constrained network interconnecting a given set of directed points in the plane and a gradient descent method for doing so in 3D space. Such a network will be referred to as a minimum Dubins tree, since its edges are Dubins paths (or slight variants thereof). The problem of constructing a minimum Dubins tree appears in the context of underground mining optimisation, where the objective is to construct a least-cost network of tunnels navigable by trucks with a minimum turning radius. The Dubins tree problem is similar to the Steiner tree problem, except the terminals are directed and there is a curvature constraint. We propose the minimum curvature-constrained Steiner point algorithm for determining the optimal location of the Steiner point in a 3-terminal network. We show that when two terminals are fixed and the third varied in the planar version of the problem, the Steiner point traces out a limaçon.

We introduce a formal framework for the study of augmenting networks in the plane for disaster-resilience, where a disaster is modeled by a straight-line segment. We generalize various graph structures from classical 2-edge-connectivity, including minimal cuts and blocks. The key concept that we introduce is that of an l-leaf, which builds on the fundamental "leaf-block " concept from classical augmentation. We present a number of algorithms for constructing the above-mentioned graph structures, including a sweep-line algorithm that finds all edge-cuts that can be destroyed by a single disaster. We also present an algorithm which optimally adds a single edge between a pair of l-leaves or blocks while avoiding certain disaster regions. Finally, we present a number of heuristic schemes for solving the disaster-resilient network augmentation problem and perform extensive experiments to demonstrate the power of the l-leaf concept within heuristic design.

Open-pit mines are surface excavations created to extract valuable material which, in most cases, is located near the surface. Pushbacks are connected regions of a mine with enough working area to support the mining operation over a defined period. Pushbacks designed without taking into consideration haulage ramps are defined here as semi-practical pushbacks. We introduce a new optimisation model for semi-practical pushbacks that accounts for operational conditions: minimum operational width and connectivity within the blocks that compose the pushbacks. Additionally, we propose an algorithm that uses a Sliding Window Heuristic, variable bounding, and other preprocessing routines to obtain solutions with less than 10% optimality gap for mining instances ranging between 30,000 to 50,000 blocks.

Consider a configuration of points comprising a point q and a set of concyclic points R that are all a given distance r from q in the Euclidean plane. In this paper, we investigate the relationship between the length of a minimum Steiner tree (MStT) on R?{q} and a minimum spanning tree on R. We show that if the degree of q in the MStT is 1, then the difference between these two lengths is at least (2-3)r, and that this lower bound is tight. This bound can be applied as part of an efficient algorithm to find the solution to the prize-collecting Euclidean Steiner tree problem, as outlined in an earlier paper.

The prize‐collecting Euclidean Steiner tree (PCEST) problem is a generalization of the well‐known Euclidean Steiner tree (EST) problem. All points given in an EST problem instance are connected by the shortest possible network in a solution. A solution can include additional points called Steiner points. A PCEST problem instance differs from an EST problem instance by the addition of weights for each given point. A PCEST solution connects a subset of the given points in order to maximize the net value of the network (the sum of the selected point weights, less than the length of the network). We present an algorithmic framework for solving the PCEST problem. Included in the framework are efficient methods to determine subsets of points that must be in every solution, and subsets of points that cannot be in any solution. Also included are methods to generate and concatenate full Steiner trees.

We present methods for simplifying the geometry of polygonal obstacles as a preprocessing step to solving obstacle-avoiding Steiner network problems in the plane. The methods reduce the total number of vertices and edges that need to be considered for the given obstacles, and their use is expected to significantly improve the efficiency of exact algorithms for solving a range of practical Steiner network problems in obstacle environments. Included are methods for extending obstacles (via a new padding method and a backfilling procedure from the literature), and various methods for simplifying obstacles, including new methods called bounding and eliminating. We show that these methods reduce the total number of obstacle vertices and edges by performing experiments on obstacles with up to 100 vertices in the presence of up to 100 terminals. The experiments utilize a modified version of a known algorithm for quickly generating large numbers of "random" polygons with hundreds of vertices. Corresponding datasets and implementations have been made available on GitHub.

In the design of open pit mines, the region to be mined is partitioned into pushbacks, subregions that allow the mining to be divided into distinct phases. Practical pushbacks are connected, satisfy a minimum width for mining equipment and include a haulage ramp. Current pushback models typically relax some or all of these mineability conditions; consequently, the outputs from those models require significant intervention by mining engineers. We present a formulation to generate maximum value practical pushbacks. A closeness factor is introduced to quantify the design's mineability. Finally, a case study of a real mine shows that our model can produce pushbacks with more practical designs and better value than traditional approaches.

A Euclidean skeleton is a set of edges in the interior (or on the boundary) of a polygon that intersects any line segment that joins two points outside of the polygon and that intersects the polygon. In this paper we study minimum cardinality Euclidean skeletons and develop an algorithm for constructing them. We first prove a number of structural properties of minimum skeletons and use these to develop a canonical form. We then design an exact algorithm which initially generates a set of canonical skeleton edges, then executes a pruning module to reduce the set of candidate edges, and finally runs existing integer linear programming code to output an optimal solution. Finally, we perform computational testing on our algorithm to demonstrate its performance, and observe a number of experimental properties of minimum skeletons.

We introduce the concept of an obstacle skeleton which is a set of line segments inside a polygonal obstacle $\omega$ that can be used in place of $\omega$ when performing intersection tests for obstacle-avoiding network problems in the plane. A skeleton can have significantly fewer line segments compared to the number of line segments in the boundary of the original obstacle, and therefore performing intersection tests on a skeleton (rather than the original obstacle) can significantly reduce the CPU time required by algorithms for computing solutions to obstacle-avoidance problems. A minimum skeleton is a skeleton with the smallest possible number of line segments. We provide an exact $O(n^2)$ algorithm for computing minimum skeletons for rectilinear obstacles in the rectilinear plane that are rectilinearly-convex. We show that the number of edges in a minimum skeleton is generally very small compared to the number of edges in the boundary of the original obstacle, by performing experiments on random rectilinearly-convex obstacles with up to 1000 vertices.

Relay node placement, which aims to connect pre-deployed sensor nodes to base stations, is essential in minimizing the costs of wireless sensor networks. In this paper, we formulate the new Node-Weighted Partial Terminal Steiner Tree Problem (NWPTSTP) for minimum-cost relay node placement in two-tiered wireless sensor networks. The objective is to minimize the sum of heterogeneous production and placement costs of relay nodes and the sum of outage probabilities of transmission routes in a routing tree simultaneously. This extends the previous work that considers the costs of relay nodes to be homogeneous. After formulating NWPTSTP for this purpose, we prove that it can be transformed to the existing node-weighted Steiner tree problem. Subsequently, we conduct some theoretical analyses on the emerging Physarum-inspired algorithms to reveal their potential of computing Steiner trees. Based on these analyses, we propose a new Physarum-inspired algorithm for solving NWPTSTP. We conduct computational trials to show that: 1) in comparison to a state-of-the-art approximation algorithm for solving the node-weighted Steiner tree problem, our Physarum-inspired algorithm can produce better solutions in a smaller amount of time; and 2) in comparison to two state-of-the-art relay node placement algorithms, our Physarum-inspired algorithm can design wireless sensor networks with 25% lower relay cost and similar quality of service (specifically, 5% shorter network lifetime, 2% longer delay, and 0% loss of goodput). This indicates the usefulness of our Physarum-inspired algorithm for minimum-cost relay node placement in budget-limited scenarios.

The problem studied in this paper is that of designing the optimal open pit haulage ramp that, for a given ramp width and a maximum ramp gradient, connects two points of the mine, minimising construction and operational costs. Because in-pit ramps require the removal of a considerable amount of non-valuable material (stripping), we discuss two different problems: high stripping (or in-pit) ramp design and low stripping (or ex-pit) road design. For the first situation, we present an integer programming model; in the second case, a shortest path approach is undertaken. In both cases, the models can be solved exactly, and include gradient and curvature constraints. The proposed formulations have been tested on real mine data, showing a significant reduction in cost compared to the previous mine design.

We present a new exact algorithm for computing minimum 2-edge-connected Steiner networks in the Euclidean plane. The algorithm is based on the GeoSteiner framework for computing minimum Steiner trees in the plane. Several new geometric and topological properties of minimum 2-edge-connected Steiner networks are developed and incorporated into the new algorithm. Comprehensive experimental results are presented to document the performance of the algorithm which can reliably compute exact solutions to randomly generated instances with up to 50 terminals-doubling the range of existing exact algorithms. Finally, we discuss the appearance of Hamiltonian cycles as solutions to the minimum 2-edge-connected Steiner network problem.

Bottleneck Steiner networks model energy consumption in wireless ad-hoc networks. The task is to design a network spanning a given set of terminals and at most $k$ Steiner points such that the length of the longest edge is minimised. The problem has been extensively studied for the case where an optimal solution is a tree in the Euclidean plane. However, in order to model a wider range of applications, including fault-tolerant networks, it is necessary to consider multi-connectivity constraints for networks embedded in more general metrics. We show that the $2$-connected bottleneck Steiner network problem is NP-hard in any planar $p$-norm and, in fact, if P$\,\neq\,$NP then an optimal solution cannot be approximated to within a ratio of ${2}^\frac{1}{p}-\epsilon$ in polynomial time for any $\epsilon >0$ and $1\leq p< \infty$.

It is challenging to design large and low-cost communication networks. In this paper, we formulate this challenge as the prize-collecting Steiner Tree Problem (PCSTP). The objective is to minimize the costs of transmission routes and the disconnected monetary or informational profits. Initially, we note that the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to improve suboptimal solutions to PCSTP. Based on these techniques, we propose two fast heuristic algorithms: the first one is a quasilinear time heuristic algorithm that is faster and consumes less memory than other algorithms; and the second one is an improvement of a state-of-the-art polynomial time heuristic algorithm that can find high-quality solutions at a speed that is only inferior to the first one. We demonstrate the competitiveness of our heuristic algorithms by comparing them with the state-of-the-art ones on the largest existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we generate new instances that are even larger (1 000 000 vertices and 10 000 000 edges) to further demonstrate their advantages in large networks. The state-of-the-art algorithms are too slow to find high-quality solutions for instances of this size, whereas our new heuristic algorithms can do this in around 6 to 45s on a personal computer. Ultimately, we apply our post-processing techniques to update the best-known solution for a notoriously difficult benchmark instance to show that they can improve near-optimal solutions to PCSTP. In conclusion, we demonstrate the usefulness of our heuristic algorithms and post-processing techniques for designing large and low-cost communication networks.

We introduce the concepts of k-lunes and k-lune inequalities, which form the basis for new geometric pruning rules for limiting the number of candidate full components that need to be considered when solving the Euclidean Steiner tree problem or the Euclidean 2-connected Steiner network problem. For the latter problem, these new pruning rules constitute the first empty region properties to have been developed for the problem. We show how to implement these rules efficiently and run computational experiments, indicating the extent to which they can improve the performance of state-of-the-art algorithms for these problems.

The stability and power sharing properties of droop-controlled inverter-based microgrids are adversely affected by model uncertainty, and inverter parameter drifts. Even when frequency stability may be guaranteed, power sharing remains sensitive to clock/frequency drifts. A novel coordinating control law is proposed to overcome these issues. It guarantees both stability and power sharing in the presence of parameter uncertainty, including frequency drift. The new control law uses sparse inter-node communications. Conditions to achieve (local) stability with power sharing are presented. It is also shown that our power sharing control is robust to reasonable clock drifts and very small droop coefficients. Simulation and experimental results illustrate the performance of the proposed control law under clock drift scenarios.

Stability and power sharing properties of droop controlled inverter-based microgrid systems depend on various design factors. Little explored is the effect of component mismatches and parameters drifts on the stability, steady state behaviour and power sharing properties of these systems. In this paper, the behaviour of frequency droop controlled inverter based microgrid systems in the presence of non-identical clocks is analysed. It is shown that power sharing between converters in a microgrid can be sensitive to clock mismatches. Our proposal shows that a coordination control that uses sparse inter-node communications is useful in ensuring desired active power sharing. Conditions are derived to ensure stability in the presence of the proposed controller and simulation results are presented.

Component mismatches and parameters drifts drastically affect stability and long-term operation of droop-controlled inverter-based microgrids. This paper analyzes and illustrates the impact of design variations and parameter drifts between angle droop controlled inverter-interfaced sources in a microgrid. It is shown thatmicrogrid stability is very sensitive to parameter drifts, especially in frequency. A coordination control scheme that uses internode communications is proposed for improving the stability margin and ensuring the desired power sharing. Conditions for stability are derived and simulation results are presented to validate the performance of the proposal.

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.