Steiner Tree Research Articles

A Framework for 5G Network Slicing Optimization using 2-Edge-Connected Subgraphs for Path Protection

Emerging telecommunications technologies require robust frameworks for efficient network slicing. We propose a network-slicing model that aims to optimize the deployment of virtual networks on a physical network topology. Our model ensures compliance with 5G requirements, incorporating latency and capacity constraints on virtual links. Selecting slices with cost and resource requirements on the computing nodes is optimized using a Knapsack problem with revenue maximization. We propose a path protection algorithm to deal with link failures by constructing a 2-edge-connected subgraph (or two link-disjoint Steiner trees) for each slice to provide both primary and backup paths. Simulation results include comparison with existing solutions by metrics such as latency, revenue, resource utilization, number of protected slices, and computation time, providing valuable insights for network planners operating in diverse and dynamic environments. Key contributions include efficient resource allocation using the Knapsack problem, enhanced network resilience via 2-edge-connected subgraphs for path protection, and realistic simulation experiments on SNDlib dataset topologies. The simulation results show that the proposed framework improves computational efficiency compared to the recent related solutions, particularly in large network topologies where k-connected function slicing (KC- FS) subgraph embeddings take approximately 3.5 times more computation time.

Uniqorn: Unified question answering over RDF knowledge graphs and natural language text

Question answering over RDF data like knowledge graphs has been greatly advanced, with a number of good systems providing crisp answers for natural language questions or telegraphic queries. Some of these systems incorporate textual sources as additional evidence for the answering process, but cannot compute answers that are present in text alone. Conversely, the IR and NLP communities have addressed QA over text, but such systems barely utilize semantic data and knowledge. This paper presents a method for complex questions that can seamlessly operate over a mixture of RDF datasets and text corpora, or individual sources, in a unified framework. Our method, called Uniqorn, builds a context graph on-the-fly, by retrieving question-relevant evidences from the RDF data and/or a text corpus, using fine-tuned BERT models. The resulting graph typically contains all question-relevant evidences but also a lot of noise. Uniqorn copes with this input by a graph algorithm for Group Steiner Trees, that identifies the best answer candidates in the context graph. Experimental results on several benchmarks of complex questions with multiple entities and relations, show that Uniqorn significantly outperforms state-of-the-art methods for heterogeneous QA – in a full training mode, as well as in zero-shot settings. The graph-based methodology provides user-interpretable evidence for the complete answering process.

Linear circuit synthesis using weighted Steiner trees

CNOT circuits are a common building block of general quantum circuits. The problem of synthesizing and optimizing such circuits has received a lot of attention in the quantum computing literature. This problem is especially challenging for quantum devices with restricted connectivity, where two-qubit gates can only be placed between adjacent qubits. The state-of-the-art algorithms for optimizing the number of CNOT gates are heuristic algorithms that are based on Gaussian elimination and that use Steiner trees to connect between different subsets of qubits. In this article, we suggest considering \emph{weighted} Steiner trees, and we present a simple low-cost heuristic to compute weights. The simulated evaluation shows that the suggested heuristic is almost always beneficial and reduces the number of CNOT gates by up to 10%.

Distributing circuits over heterogeneous, modular quantum computing network architectures

We consider a heterogeneous network of quantum computing modules, sparsely connected via Bell states. Operations across these connections constitute a computational bottleneck and they are likely to add more noise to the computation than operations performed within a module. We introduce several techniques for transforming a given quantum circuit into one implementable on such a network, minimising the number of Bell states required to do so. We extend previous works on circuit distribution to the case of heterogeneous networks. On the one hand, we extend the hypergraph approach of Andres-Martinez and Heunen (2019 Phys. Rev. A 100 032308) to arbitrary network topologies, and we propose the use of Steiner trees to detect and reuse common connections, further reducing the cost of entanglement sharing within the network. On the other hand, we extend the embedding techniques of Wu et al (2023 Quantum 7 1196) to networks with more than two modules. We show that, with careful manipulation of trade-offs, these two new approaches can be combined into a single automated framework. Our proposal is implemented and benchmarked; the results confirm that our contributions make noticeable improvements upon the aforementioned works and complement their weaknesses.

Scattering and Sparse Partitions, and Their Applications

A partition \(\mathcal{P}\) of a weighted graph \(G\) is \((\sigma,\tau,\Delta)\) -sparse if every cluster has diameter at most \(\Delta\) , and every ball of radius \(\Delta/\sigma\) intersects at most \(\tau\) clusters. Similarly, \(\mathcal{P}\) is \((\sigma,\tau,\Delta)\) -scattering if instead for balls, we require that every shortest path of length at most \(\Delta/\sigma\) intersects at most \(\tau\) clusters. Given a graph \(G\) that admits a \((\sigma,\tau,\Delta)\) -sparse partition for all \(\Delta > 0\) , Jia et al. constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch \(O(\tau\sigma^{2} \log _{\tau}n)\) . Given a graph \(G\) that admits a \((\sigma,\tau,\Delta)\) -scattering partition for all \(\Delta > 0\) , we construct a solution for the Steiner Point Removal problem with stretch \(O(\tau^{3}\sigma^{3})\) . We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.

Streaming Algorithms for Geometric Steiner Forest

We consider a generalization of the Steiner tree problem, the Steiner forest problem , in the Euclidean plane: the input is a multiset \(X\subseteq{\mathbb{R}}^{2}\) , partitioned into \(k\) color classes \(C_{1},\ldots,C_{k}\subseteq X\) . The goal is to find a minimum-cost Euclidean graph \(G\) such that every color class \(C_{i}\) is connected in \(G\) . We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to \(X\) . Each input point \(x {\in} X\) arrives with its color \(\mathsf{color}(x) {\in} [k]\) , and as usual for dynamic geometric streams, the input is restricted to the discrete grid \(\{1,\ldots,\Delta\}^{2}\) . We design a single-pass streaming algorithm that uses \(\operatorname{poly}(k\cdot\log\Delta)\) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio \(\alpha_{2}\) (currently \(1.1547\leq\alpha_{2}\leq 1.214\) ). This approximation guarantee matches the state-of-the-art bound for streaming Steiner tree, i.e., when \(k=1\) , and it is a major open question to improve the ratio to \(1+\varepsilon\) even for this special case. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and so far has not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite multiplicative approximation requires \(\Omega(k)\) bits of space.

Polynomial Integrality Gap of Flow LP for Directed Steiner Tree

In the Directed Steiner Tree (DST) problem, we are given a directed graph \(G=(V,E)\) on \(n\) vertices with edge-costs \(c\in{\mathbb{R}}_{\geq 0}^{E}\) , a root vertex \(r\in V\) , and a set \(K\subseteq V\setminus\{r\}\) of \(k\) terminals. The goal is to find a minimum-cost subgraph of \(G\) that contains a path from \(r\) to every terminal \(t\in K\) . DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of \(O(k^{\epsilon})\) for any constant \(\epsilon {\gt} 0\) , and the best quasi-polynomial-time approximation ratio of \(O\left(\frac{\log^{2}k}{\log\log k}\right)\) . Towards understanding this gap, we study the integrality gap of the standard flow linear programming relaxation for the problem. We show that the linear program (LP) has an integrality gap of \(\Omega(n^{0.0418})\) . Previously, the integrality gap of the LP is only known to be \(\Omega\left(\frac{\log^{2}n}{\log\log n}\right)\) [Halperin et al., SODA’03 & SIAM J. Comput.] and \(\Omega(\sqrt{k})\) [Zosin-Khuller, SODA’02] in some instance with \(\sqrt{k}=O\left(\frac{\log n}{\log\log n}\right)\) . Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in \(n\) , the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.

A hybrid genetic algorithm for the steiner tree problem in graphs towards optimizations for wireless sensor networks

A hybrid genetic algorithm for the steiner tree problem in graphs towards optimizations for wireless sensor networks

A heuristic method for solving the Steiner tree problem in graphs using network centralities

We propose a heuristic method of using network centralities for constructing small-weight Steiner trees in this paper. The Steiner tree problem in graphs is one of the practical NP-hard combinatorial optimization problems. Given a graph and a set of vertices called terminals in the graph, the objective of the Steiner tree problem in graphs is to find a minimum weight Steiner tree that is a tree containing all the terminals. Conventional construction methods make a Steiner tree based on the shortest paths between terminals. If these shortest paths are overlapped as much as possible, we can obtain a small-weight Steiner tree. Therefore, we proposed to use network centralities to distinguish which edges should be included to make a small-weight Steiner tree. Experimental results revealed that using the vertex or the edge betweenness centralities contributes to making small-weight Steiner trees.

New Algorithms for Steiner Tree Reoptimization

Reoptimization is a setting in which we are given a good approximate solution of an optimization problem instance and a local modification that slightly changes the instance. The main goal is that of finding a good approximate solution of the modified instance. We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {NP}$$\\end{document}-hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decades. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P=NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {P}=\ extsf {NP}$$\\end{document}

A methodology for feasibility analyses of district heating networks: A case study applied to greenhouse crops

In this article, the preliminary routing, sizing, and cost structure of a heat network that supplies energy, in an agrarian district, to greenhouses located around a cogeneration plant in the municipality of El Ejido (Almería) is presented. A genetic algorithm is employed to select a set of points of consumption (plots of land)–which indirectly determine the network routing over the road network–resulting in a project with a better net present value. To obtain the routing, the Steiner tree problem is solved through an approximation that involves several algorithms derived from graph theory and the use of georeferenced information. The design is constrained to ensure technical feasibility, so that the available thermal power from the plant, averaged for hourly periods over a year, is greater than the demand from the greenhouses. The said demand is calculated through an empty-greenhouse climate model, considering temperature setpoints compatible with the most common crops in the region. According to the results, a network of total length equal to 1.3464 km and an average pipe diameter of 0.1532 m would be required to annually supply 8621.4 MWh of heat to 12 points of consumption. The net present value of the project would be 6 417 900 €, considering a 3% discount rate and useful life of 30 years, with an initial investment cost of 1 723 300 € and an annual operation cost of 15 800 €year−1. A sensibility analysis of the variables with greater economic impact is also presented.

An online transfer learning based multifactorial evolutionary algorithm for solving the clustered Steiner tree problem

The rapid increase in the number and computing power of devices interacting through the network over the past few years has led to a growing trend of decentralized clustering network architecture. The Clustered Steiner Tree Problem (CluSteiner) is a recently introduced NP-Hard problem and is assessed to be the core of efficient multicast routing in such networks. However, research on this problem in terms of theory and algorithmic design is still in its infancy as current solving approaches are limited in the literature. Lately, the Multifactorial Evolutionary Algorithm (MFEA) has been applied to solve several clustering-structure network design problems for the reason that these problems rarely exist independently and are often deployed concurrently in practice. To effectively transfer knowledge between problems while solving them simultaneously, this paper proposes an approach based on the novel data-driven MFEA-II to deal with the CluSteiner problem. In the proposal, an efficient encoding and a two-level decoding mechanism are introduced, combined with a module capable of online learning the amount of knowledge transferred between tasks that help the algorithm to explore potential regions of the search space. Experiments are undertaken on various test instances to evaluate the efficacy of the proposed algorithm. The empirical results indicate that our proposal can perform well on both metric and non-metric graphs with the statistical tests used to verify its efficiency over other baseline algorithms regarding the solution quality and convergence trend.

An efficient solver for large-scale onshore wind farm siting including cable routing

Existing planning approaches for onshore wind farm siting and grid integration often do not meet minimum cost solutions or social and environmental considerations. In this paper, we develop an exact approach for the integrated layout and cable routing problem of onshore wind farm planning using the Quota Steiner tree problem. Applying a novel transformation on a known directed cut formulation, reduction techniques, and heuristics, we design an exact solver that makes large problem instances solvable and outperforms generic MIP solvers. In selected regions of Germany, the trade-offs between minimizing costs and landscape impact of onshore wind farm siting are investigated. Although our case studies show large trade-offs between the objective criteria of cost and landscape impact, small burdens on one criterion can significantly improve the other criteria. In addition, we demonstrate that contrary to many approaches for exclusive turbine siting, grid integration must be simultaneously optimized to avoid excessive costs or landscape impacts in the course of a wind farm project. Our novel problem formulation and the developed solver can assist planners in decision-making and help optimize wind farms in large regions in the future.

An Optimized Data Storage in A Secure Cloud-Edge Collaboration A Fault Tolerance Approach

The rise of edge smart IoT devices has led to the development of edge storage systems (ESS) for efficient access to massive edge data. ESS can reduce the load on cloud centers and improve user experience. However, ESS still faces challenges in improving fault tolerance and efficiency. Thus, there is a need for a secure and efficient fault-tolerant storage scheme. Existing schemes have drawbacks like high edge storage overhead, difficulty in protecting edge data privacy, and low data writing performance. To address these issues, we propose a Hierarchical Cloud-Edge Collaborative Fault-Tolerant Storage (HCEFT) model. This model aims to enhance system robustness, reduce edge storage overhead, and ensure edge data privacy. We also introduce an optimization method for data writing in HCEFT, called ECWSS (Erasure Code data Writing method based on Steiner tree and SDN). This method improves the trade-off between data writing time and traffic consumption. Our scheme improves data robustness, availability, and security. Additionally, the writing optimization method reduces data write time by 13%-67% and network traffic consumption by 20%-62%, enhancing network load balance performance.

Parameterized algorithms for Steiner tree and (connected) dominating set on path graphs

AbstractChordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).

NN-Steiner: A Mixed Neural-Algorithmic Approach for the Rectilinear Steiner Minimum Tree Problem

Recent years have witnessed rapid advances in the use of neural networks to solve combinatorial optimization problems. Nevertheless, designing the "right" neural model that can effectively handle a given optimization problem can be challenging, and often there is no theoretical understanding or justification of the resulting neural model. In this paper, we focus on the rectilinear Steiner minimum tree (RSMT) problem, which is of critical importance in IC layout design and as a result has attracted numerous heuristic approaches in the VLSI literature. Our contributions are two-fold. On the methodology front, we propose NN-Steiner which is a novel mixed neural-algorithmic framework for computing RSMTs that leverages the celebrated PTAS algorithmic framework of Arora to solve this problem (and other geometric optimization problems). Our NN-Steiner replaces key algorithmic components within Arora's PTAS by suitable neural components. In particular, NN-Steiner only needs four neural network (NN) components that are called repeatedly within an algorithmic framework. Crucially, each of the four NN components is only of bounded size independent of input size, and thus easy to train. Furthermore, as the NN component is learning a generic algorithmic step, once learned, the resulting mixed neural-algorithmic framework generalizes to much larger instances not seen in training. Our NN-Steiner, to our best knowledge, is the first neural architecture of bounded size that has capacity to approximately solve RSMT (and variants). On the empirical front, we show how NN-Steiner can be implemented and demonstrate the effectiveness of our resulting approach, especially in terms of generalization, by comparing with state-of-the-art methods (both neural and non-neural based).

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.

Mixed Integer Programming based Placement Refinement by RSMT Model with Movable Pins

Placement is a critical step in the physical design for digital application specific integrated circuits (ASICs), as it can directly affect the design qualities such as wirelength and timing. For many domain specific designs, the demands for high performance parallel computing result in repetitive hardware instances, such as the processing elements in the neural network accelerators. As these instances can dominate the area of the designs, the runtime of the complete design’s placement can be traded for optimizing and reusing one instance’s placement to achieve higher quality. Therefore, this work proposes a mixed integer programming (MIP)-based placement refinement algorithm for the repetitive instances. By efficiently modeling the rectilinear steiner tree wirelength, the placement can be precisely refined for better quality. Besides, the MIP formulations for timing-driven placement are proposed. A theoretical proof is then provided to show the correctness of the proposed wirelength model. For the instances in various popular fields, the experiments show that given the placement from the commercial placers, the proposed algorithm can perform further placement refinement to reduce 3.76%/3.64% detailed routing wirelength and 1.68%/2.42% critical path delay under wirelength/timing-driven mode, respectively, and also outperforms the state-of-the-art previous work.

On convexity in split graphs: complexity of Steiner tree and domination

On convexity in split graphs: complexity of Steiner tree and domination

Flow-network adaptation and behavior in slime molds

The slime mold Physarum polycephalum is an amoebozoa that grows forming a cytoplasm network that adapts its geometry to external stimuli. The cytoplasm is made of ectoplasm tubes in which the endoplasmic fluid flows. Endoplasmic flow is due to the rhythmic contraction of the actomyosin fibers of the ectoplasm, which induces a peristaltic wave that can be tracked through the spatiotemporal variations of the tube diameters. Slime mold behavior depends on many periodic modes of tube diameter variation, which is believed to allow a smooth transition between migration directions. Physarum polycephalum can solve mazes and grow optimal networks to solve traveling salesman and Steiner tree problems. Slime mold network dynamics have been modeled through cell automata and stochastic approaches, as well as fluid flow equations, electronic analogs, and multi-agent systems. Here, we examine the modeling strategies available to date to simulate flow-network adaptation in slime molds. However, we found no theoretical framework that can properly predict the evolution of the network as it morphs from an initial configuration to a pseudo-asymptotic optimum or explain the physical phenomena that drive endoplasmic flow or memory encoding at the scale of the entire network. Multi-frame object tracking by k-partite graphs holds promise for slime mold network analysis and tracking, whereas deep learning could be used to classify sequences of latent features to help characterize the behavior of Physarum polycephalum. The combination of the two could pave the way to a new class of predictive behavior models for slime molds.