Steiner Tree Research Articles

Determining the Minimum Cost Steiner Tree for Delay Constrained Problems

We address a variant of the Steiner tree problem for delay constrained problems. The addressed problem consists in determining the minimum cost Steiner tree, while guaranteeing that the delay between any two terminal nodes does not exceed a given maximum value. This problem is known as the bounded diameter Steiner minimum tree problem. We propose a compact formulation based on integer linear programming (ILP) to obtain optimal solutions, which was efficiently solved on two telecommunication core networks up to 75 nodes. However, given that for traditional Steiner tree graphs the ILP proved to be inefficient, we propose a heuristic method and compare it with the ILP formulation. We show that the heuristic provides optimal solutions, except for two cases in our experiments where it provided near-optimal solutions, always in reasonable runtimes. Additionally, to reduce the complexity of the problem, we propose some novel and modified graph reductions specific for the addressed problem.

Using multiflow formulations to solve the Steiner tree problem in graphs

We present three different mixed integer linear models with a polynomial number of variables and constraints for the Steiner tree problem in graphs. The linear relaxations of these models are compared to show that a good (strong) linear relaxation can be a good approximation for the problem. We present computational results for the STP OR-Library (J.E. Beasley) instances of type b, c, d and e.

Solving the Delay-Constrained Least-Cost routing problem using Tabu Search with Edge Betweenness

Multicast routing consists of concurrently sending the same information from a source to a subset of all possible destinations in a computer network thus becomes an important technology communication. To solve the problem, a current approach for efficiently supporting a multicast session in a network consists of establishing a multicast tree that covers the source and all terminal nodes. This problem can be reduced to a minimal Steiner tree problem (MST) which aims to look for a tree that covers a set of nodes with a minimum total cost, the problem is NP-hard. In this paper, we investigate metaheuristics approaches for the Delay-Constrained Least-Cost (DCLC) problem, we propose a novel algorithm based on Tabu Search procedure with the Edge Betweenness (EB). The EB heuristic used first to improve KMB heuristic, able to measure the edge value to being included in a given path. The obtained solution improved using the tabu search method. The performance of the proposed algorithm is evaluated by experiments on a number of benchmark instances from the Steiner library. Experimental results show that the proposed metaheuristic gives competitive results in terms of cost and delay compared to the optimal results in Steiner library and other existing algorithms in the literature.

A New Method for Reconstructing Data on a Single Failure Node in the Distributed Storage System Based on the MSR Code

As a storage method for a distributed storage system, an erasure code can save storage space and repair the data of failed nodes. However, most studies that discuss the repair of fault nodes in the erasure code mode only focus on the condition that the bandwidth of heterogeneous links restricts the repair rate but ignore the condition that the storage node is heterogeneous, the cost of repair traffic in the repair process, and the influence of the failure of secondary nodes on the repair process. An optimal repair strategy based on the minimum storage regenerative (MSR) code and a hybrid genetic algorithm is proposed for single‐node fault scenarios to solve the above problems. In this work, the single‐node data repair problem is modeled as an optimization problem of an optimal Steiner tree with constraints considering heterogeneous link bandwidth and heterogeneous node processing capacity and takes repair traffic and repair delay as optimization objectives. After that, a hybrid genetic algorithm is designed to solve the problem. The experimental results show that under the same scales used in the MSR code cases, our approach has good robustness and its repair delay decreases by 10% and 55% compared with the conventional tree repair topology and star repair topology, respectively; the repair flow increases by 10% compared with the star topology, and the flow rate of the conventional tree repair topology decreases by 40%.

Cost-Efficient Scheduling of Multicast Transfers with Deadline Guarantees Across Edge Datacenters

Moderate-scale datacenters are increasingly deployed at the network edge to support low-latency and high-bandwidth internet of Things (IoT) and 5G applications. These applications usually have bulk data to transfer from one datacenter to several datacenters, which can cause a significant amount of bandwidth costs. Traffic engineering (TE) systems at the network edge must minimize the bandwidth costs of multicast transfers across edge datacenters. To avoid service quality degradation, TE should also guarantee transfer deadlines, which have received little attention in existing work. Due to the dynamic bandwidth pricing schemes, it is challenging to strike a good balance between minimizing costs and guaranteeing transfer deadlines. In this paper, we present CDScheduler a cost-efficient scheduling solution for multicast transfers with deadline guarantees. To reduce bandwidth costs, CDScheduler uses Steiner trees for forwarding and adaptive routing that considers bandwidth price variation and transfer demands. We formulate the cost-efficient multicast transfer scheduling problem and propose an algorithm based on linear program relaxation and randomized rounding to find a solution that guarantees timely completion and reduces cost. Extensive evaluations show that CDScheduler can reduce bandwidth cost significantly and outperforms the state-of-the-art solutions by cutting down up to 78% bandwidth cost.

Chvátal–Gomory cuts for the Steiner tree problem

The Steiner tree problem asks for a minimum weight subtree of an undirected graph spanning distinct terminal nodes. As one of the most fundamental network design problems, it has received much attention from both the mathematical programming and the algorithmic community and indeed, many integer linear programming (ILP) formulations of varying strength are known. The practical usefulness of the formulations can be enhanced with additional valid inequalities that reduce the gap between the efficiently computable continuous relaxation and the computationally difficult integer linear program. We aim to better understand the structure of these inequalities. In particular, we want to know how the inequalities can be obtained through Chvátal–Gomory cuts; a general procedure that is implemented in state-of-the-art ILP solver software. We start with the most basic ILP formulation for the Steiner tree problem and analyze how two known classes of strengthening inequalities – partition inequalities and odd-hole inequalities – can be derived systematically as Chvátal–Gomory cuts and how these inequalities themselves can be used to derive Chvátal–Gomory cuts. Along the way, we prove that the Chvátal rank of k-partition and k-odd-hole inequalities is at most k−2 and k−1, respectively. In particular, this proves that k iterations of the Chvátal–Gomory cut procedure suffice to obtain a given k+2-partition or a given k+1-odd-hole inequality.

Mixed Integer Programming Formulations for Steiner Tree and Quality of Service Multicast Tree Problems

This article presents flow-based mixed integer programming formulations for the Steiner Tree Problem and its variant applied to model and solve the Quality of Service Multicast Tree problem. This is a relevant problem related to nowadays telecommunication networks, particularly Content Delivery Networks, to distribute multimedia over cloud-based Internet systems. To the best of our knowledge, no previous mixed integer programming formulation was proposed for the Quality of Service Multicast Tree problem variant. Experimental evaluation is performed over a set of realistic problem instances from SteinLib, a reference test-set repository, modified accordingly to prove that standard exact solvers are capable of finding solutions to real-world size instances. Exact methods are applied for benchmarking the proposed formulations, finding optimal solutions and low feasible-to-optimal gaps in reasonable execution times.

Solving Steiner trees: Recent advances, challenges, and perspectives

AbstractThe Steiner tree problem (STP) in graphs is one of the most studied problems in combinatorial optimization. Since its inception in 1970, numerous articles published in the journal Networks have stimulated new theoretical and computational studies on Steiner trees: from approximation algorithms, heuristics, metaheuristics, all the way to exact algorithms based on (mixed) integer linear programming, fixed parameter tractability, or combinatorial branch‐and‐bounds. The pervasive applicability and relevance of Steiner trees have been reinforced by the recent 11th DIMACS Implementation Challenge in 2014 and the PACE 2018 Challenge. This article provides an overview of the rich developments from the last three decades for the STP in graphs and highlights the most recent computational studies for some of its closely related variants.

To cut or to fill

We present a novel algorithm for simplifying the topology of a 3D shape, which is characterized by the number of connected components, handles, and cavities. Existing methods either limit their modifications to be only cutting or only filling, or take a heuristic approach to decide where to cut or fill. We consider the problem of finding a globally optimal set of cuts and fills that achieve the simplest topology while minimizing geometric changes. We show that the problem can be formulated as graph labelling, and we solve it by a transformation to the Node-Weighted Steiner Tree problem. When tested on examples with varying levels of topological complexity, the algorithm shows notable improvement over existing simplification methods in both topological simplicity and geometric distortions.

Towards the Sustainable Development Goals: A Bi-objective framework for electricity access

Traditionally, the main focus of evaluation in universal electricity access problems has been cost. However, additional criteria such as increasing renewable penetration due to environmental concerns or grid penetration due to reliability concerns, have become increasingly important. We acknowledge the importance of additional criteria and propose a bi-objective framework so as to help decision makers investigate the trade-offs between potentially conflicting criteria in rural electrification. We consider two objective space based exact approaches using the Prize Collecting Steiner Tree (PCST) formulation and two metaheuristic algorithms to find Pareto solutions, and investigate their performances on real life problem instances. This study is expected to be an important decision support tool for the electrification of underdeveloped communities, having the potential of contributing to their socio-economic development.

Efficient VLSI routing algorithm employing novel discrete PSO and multi-stage transformation

For routing industrial circuits, the Steiner minimal tree (SMT) model can be applied in different routing problems, such as wirelength optimization, congestion reduction, and delay optimization. In this paper, an efficient VLSI routing algorithm employing novel discrete particle swarm optimization (PSO) and multi-stage transformation is proposed to build two types of SMT, including X-architecture Steiner minimal tree and rectilinear Steiner minimal tree. Firstly, to simultaneously handle two types of SMT problems, an effective encoding strategy is proposed to be more suitable for PSO and thus it can overcome the difficulty of designing different algorithms for different routing architectures. Secondly, a multi-stage transformation strategy is presented to expand the search space of the proposed algorithm and accelerate the convergence speed of the proposed algorithm. Various combinations of multi-stage transformation strategies have been tested to highlight the best combination. Furthermore, the various genetic operators combined with union-find data structure strategy are proposed to construct the novel and effective discrete particle update formula. Experimental simulation results on industrial circuits show that the proposed algorithm can get the best solutions among the existing algorithms.

Urban Free-Space Optical Network Optimization

This paper presents a set of graph optimization problems related to free-space optical communication networks. Such laser-based wireless networks require a line of sight to enable communication, thus a visibility graph model is used herein. The main objective is to provide connectivity from a communication source point to terminal points through the use of some subset of available intermediate points. To this end, we define a handful of problems that differ mainly in the costs applied to the nodes and/or edges of the graph. These problems should be optimized with respect to cost and performance. The problems at hand are shown to be NP-hard. A generic heuristic based on a genetic algorithm is proposed, followed by a set of simulation experiments that demonstrate the performance of the suggested heuristic method on real-life scenarios. The suggested genetic algorithm is compared with the Euclidean Steiner tree method. Our simulations show that in many settings, especially in dense graphs, the genetic algorithm finds lower-cost solutions than its competitor, while it falls behind in some settings. However, the run-time performance of the genetic algorithm is considerably better in graphs with 1000 nodes or more, being more than twice faster in some settings. We conclude that the suggested heuristic improves run-time performance on large-scale graphs and can handle a wider range of related optimization problems. The simulation results suggest that the 5G urban backbone may benefit significantly from using free-space optical networks.

Distributed Approximation Algorithms for Steiner Tree in the CONGESTED CLIQUE

The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGESTED CLIQUE model (CCM) [29] of distributed computing. For the Steiner tree problem in the CCM, we consider that each vertex of the input graph is uniquely mapped to a processor and edges are naturally mapped to the links between the corresponding processors. Regarding output, each processor should know whether the vertex assigned to it is in the solution or not and which of its incident edges are in the solution. We present two deterministic distributed approximation algorithms for the Steiner tree problem in the CCM. The first algorithm computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages for a given connected undirected weighted graph of [Formula: see text] nodes. Note here that [Formula: see text] notation hides polylogarithmic factors in [Formula: see text]. The second one computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] and [Formula: see text] are the shortest path diameter and number of edges respectively in the given input graph. Both the algorithms achieve an approximation ratio of [Formula: see text], where [Formula: see text] is the number of leaf nodes in the optimal Steiner tree. For graphs with [Formula: see text], the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with [Formula: see text], the second algorithm outperforms the first one in terms of the round complexity. In fact when [Formula: see text] then the second algorithm achieves a round complexity of [Formula: see text] and message complexity of [Formula: see text]. To the best of our knowledge, this is the first work to study the Steiner tree problem in the CCM.

A Heuristic for the Weighted Steiner Tree Problem by Using Random Delaunay Networks

本研究は重み付きシュタイナー問題の発見的解法を開発する．重み付きシュタイナー問題は平面上の与点を連結するグラフの辺の総コストを最小化する問題である．本研究で構築する解法は，ランダムドロネー網を用いて連続平面を離散化し，与点のボロノイ図の双対グラフに含まれる全域木を探索する．木の辺は，ランダムドロネー網上の重み付き最短路で与えられる．近似解の形状と解の総コストの観点から，本手法が既往の重み付きシュタイナー木の発見的解法より厳密解に近い解が得られることを示す．さらに，重みの変化に対する木の形状の不連続な変化を把握する．最後に本手法を，新たな旅客，貨物輸送手段として期待されている大型ドローンの航空路網に適用し，本手法の実用性を検証する．

Weighted amplifiers and inapproximability results for Travelling Salesman problem

The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from frac{123}{122} (Karpinski et al. in J Comput Syst Sci 81(8):1665–1677, 2015) to frac{117}{116}. This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems.

Non-Greedy Online Steiner Trees on Outerplanar Graphs

This paper addresses the classic online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we do not know any greedy or non-greedy tight deterministic algorithm for other classes of graphs. In this paper, we observe that a greedy algorithm is $$\Omega (\log n)$$ -competitive on outerplanar graphs, where n is the number of vertices, and propose a 5.828-competitive deterministic algorithm on outerplanar graphs. Our algorithm connects a requested vertex and the tree constructed thus far using a path that is constant times longer than the distance between them. We also present a lower bound of 4 for arbitrary deterministic online Steiner tree algorithms on outerplanar graphs.

Primal-dual based distributed approximation algorithm for Prize-collecting Steiner tree

The Prize-collecting Steiner tree (PCST) problem is a generalization of the Steiner tree problem that finds applications in network design, content distribution networks, and many more. There are a few centralized approximation algorithms [D. Bienstock, M. X. Goemans, D. Simchi-Levi and D. Williamson, A note on the prize collecting traveling salesman problem. Math. Program. 59 (1993) 413–420; M. X. Goemans and D. E. Williamson, A general approximation technique for constrained forest problems, SIAM J. Appl. Math. 24(2) (1995) 296–317; D. S. Johnson, M. Minkoff and S. Phillips, The prize collecting Steiner tree problem: Theory and practice, in Proc. Eleventh Annual ACM-SIAM Symp. Discrete Algorithms, SODA ’00 (2000), pp. 760–769; A. Archer, M. Hossein Bateni and M. Taghi Hajiaghayi, Improved approximation algorithms for prize-collecting Steiner tree and TSP, SIAM J. Comput. 40(2) (2011) 309–332] for solving the PCST problem. However, the problem has seen very little progress in the distributed setting; to the best of our knowledge, the only distributed algorithms proposed so far are due to Rossetti [N. G. Rossetti, A first attempt on the distributed prize-collecting Steiner tree problem, M.Sc. thesis, University of Iceland, Reykjavik (2015)]: one of them fails to guarantee a constant approximation factor while the other one is essentially centralized. In this work, first, we present a deterministic [Formula: see text] factor distributed approximation algorithm (D-PCST algorithm) that constructs a PCST for a given connected undirected graph of [Formula: see text] nodes with non-negative edge weights and non-negative prize value for each node. The D-PCST algorithm is based on the primal-dual method and uses a technique of preserving dual constraints in a distributed manner, without relying on knowledge of the global structure of the network. For an input graph [Formula: see text], the round and message complexities of the D-PCST algorithm in the CONGEST model are [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] and [Formula: see text]. Furthermore, we modify the D-PCST algorithm and show that a [Formula: see text]-approximate PCST can be deterministically computed using [Formula: see text] rounds and [Formula: see text] messages in the CONGEST model, where [Formula: see text] is the unweighted diameter of [Formula: see text]. For networks with [Formula: see text], the modified D-PCST algorithm performs better than the original one in terms of the round complexity. Both the algorithms require [Formula: see text] bits of memory in each node, where [Formula: see text] is the maximum degree of a node in the graph.

EASBVN: efficient approximation scheme for broadcasting in vehicular networks

Vehicular ad-hoc networks (VANETs) suffer from dis-connectivity due to very high mobility, node sparseness, and lossy link. Thus, the broadcasting algorithms in the vehicular multi-hop scenario have to operate in all extreme scenarios as dense, sparse, and disconnected topologies that are changeable dynamically due to different traffic conditions. The store-and-carry-forward scheme for disconnected topologies with dynamic nodes can be model as a Steiner-tree problem due to link weight change dynamically. The Steiner tree has to be recomputed to obtain the new Steiner tree, making it more difficult for broadcasting algorithms in VANETs. We show that broadcasting in VANETs (i.e., given graph $$G(V_1, E_1)$$ with the terminal set, $$T_1 \subseteq V1$$ ), modeled as to find as many (maximum) disjoint Steiner trees as possible that are disjoint on the non-terminal nodes and the edges are APX-hard even for small $$T_1$$ . If all the nodes in G are terminal nodes, i.e., $$T_1=V_1$$ , the trouble is finding a maximum cardinality set of edge-disjoint spanning trees and encountering a considerable lot of edge-disjoint spanning tree. We then propose an $$O(\log n)$$ -Approximation Algorithm for Broadcasting in VANETs via bipartite graph ( $$G=((T_1,V_1/T_1),E_1)$$ ) and using projection. We compared the performance of the proposed scheme (i.e., EASBVN) with existing state-of-the-scheme and found it to be better than the state-of-the-art schemes (i.e., DV-CAST, UVCAST, DPPEMB, MoZo, and MobiVNDN) in terms of end-to-end delay and re-healing time.

Improving acceptance rate of QoS-guaranteed point-to-multipoint traffic flows in elastic optical networks

Abstract This paper studies the load-balanced routing and admission control for point-to-multipoint traffic flows in elastic optical networks (EONs). EONs have emerged as a promising technology to provide high capacity for next-generation networks. Elastic bandwidth allocation promotes the spectrum utilization efficiency. We consider the problem of constructing bandwidth-guaranteed trees for on-demand point-to-multipoint session requests. A request is accepted, if a multicast tree with sufficient bandwidth on each link can be established. The main objective is to increase the multicast acceptance rate in the network. For this purpose, we propose an efficient scheme, namely optical load-balanced multicast routing with admission control (OLMR-AC), to improve the wavelength resources utilization in the network. As an advantage, OLMR-AC jointly considers both distance and congestion factors to create a multicast session. For each admitted session, a bandwidth–guaranteed steiner tree is provided in which not only the source node has a minimum-cost path to each receiver, but also its overall cost is minimal. The proposed scheme distributes the on-demand traffic flows over the available links fairly. Simulation results show the efficiency of the OLMR-AC algorithm in terms of multicast acceptance rate, average length of lightpaths, throughput, number of transmissions, and load variance.

Dynamic VNF Forwarding Graph Extension Algorithms

This paper proposes a set of algorithms to extend initially deployed Virtualized Network Functions Forwarding Graphs (VNF-FG or Service Function Chains - SFCs) previously requested and acquired by tenants. The latter are likely to request extension of their already acquired, deployed, and running dedicated slices to embed additional networking functions to handle for instance new application flows requiring additional classification, authentication, authorization, processing, traffic steering, etc. The tenants (or end-user service providers) are also likely to introduce gradually their services as needs arise and as their consumer base and profiles evolve. Increasing traffic load often will also require extension of the service graphs and chaining such as introducing additional load balancers and servers, new security functions, and simply new VNFs related to new tenant services. The related service graph extension problem is addressed first through an ILP algorithm that serves as a reference for performance comparisons with a set of proposed heuristic algorithms. A heuristic directly inspired by and based on the ILP and the proposed alternative approaches are shown to scale much better than the ILP while providing good solutions. The solutions include an adaptation of an eigendecomposition approach that provides very good performance for highly connected graphs. The problem is also cast into the well know Steiner Tree search problem whose complexity and performance are formally characterized in the literature. The Steiner based algorithm is shown to provide the best overall tradeoff (in rejection rate, complexity, and quality) with performance closest to optimal.