Minimum-cost Tree 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.

A multipopulation multitasking evolutionary scheme with adaptive knowledge transfer to solve the clustered minimum routing cost tree problem

The advancement in communication technology nowadays has increased the number of networked devices exponentially. Thus, network applications begin to cluster terminals to address scalability, privacy, and device connectivity with low routing costs. A recently introduced NP-hard problem called the Clustered Minimum Routing Cost Tree (CluMRCT) problem is assessed to be a fundamental key in designing an efficient network with low routing costs. Lately, some evolutionary multitasking meta-heuristics have been proposed to tackle multiple CluMRCT problems concurrently. However, these previous studies still have some limitations, such as only being applicable to complete graphs or there is no mechanism for effective knowledge transfer between problems when solving them simultaneously. To overcome these limitations, this paper proposes a novel multi-population multitasking evolutionary framework to address the problem. The individual representation is developed based on Network random key scheme and can be applied to both sparse and complete graphs. Moreover, our proposal can adaptively exploit positive knowledge transfer by adjusting the number of individuals migrating from other populations to a particular task. Then, the proposed algorithm is experimented on various data types and compared with other state-of-the-art algorithms. The experimental results show the proposal's effectiveness in most cases of the dataset, followed by detailed comparison, evaluation, and analysis.

Truthful Mechanisms for Steiner Tree Problems

Consider an undirected graph G=(V,E) model for a communication network, where each edge is owned by a selfish agent, who reports the cost for offering the use of her edge. Note that each edge agent may misreport her own cost for the use of the edge for her own benefit. In such a non-cooperative setting, we aim at designing an approximately truthful mechanism for establishing a Steiner tree, a minimum cost tree spanning over all the terminals. We present a truthful-in-expectation mechanism that achieves the approximation ratio ln 4 + ε ≈ 1.39, which matches the current best algorithmic ratio for STP.

Minimum weight clustered dominating tree problem

We discuss minimum weight clustered dominating trees that find applications in the wireless sensor network design based on a clustered independent set structure. A cluster consists of a master sensor and the sensors belonging to its sensing radius. Masters collect, filter, and transmit the sensed data to a central sensor responsible for processing all sensed information. The data sent from a cluster to the central sensor follow a unique path. It alternates between a master and a bridge node, in this order. A bridge allows data communication between two neighboring clusters. The larger the distance between masters and bridges, the higher the energy consumption for data transmission. To reduce energy consumption and increase the network lifetime, we investigate a clustered tree structure of minimum total link distances. We propose hop- and flow-based models, introduce valid inequalities for them, and discuss five exponential families of cuts when embedded into the branch-and-cut framework of the CPLEX solver. Our models benefit properly of the CPLEX Benders’ decomposition. We also highlight the differences between the topology of the clustered tree of minimum cost and the one of the (non-clustered) minimum dominating tree of the corresponding instances in terms of cost and number of solution nodes.

Steiner Tree Problem in Graphs and Mixed Integer Linear Programming-Based Approach in GAMS

The Steiner tree problem in graphs involves finding a minimum cost tree which connects a defined subset of the vertices. This problem generalises the minimum spanning tree problem, in contrast, it is NP-complete and is usually solved for large instances by deterministic or stochastic heuristic methods and approximate algorithms. In this paper, however, we focus on a different approach, based on the formulation of a mixed integer programming model and its modification for solving in the professional optimization tool GAMS, which is now capable of solving even large instances of problems of exponential complexity.

On Hop-Constrained Steiner Trees in Tree-Like Metrics

We consider the problem of computing a Steiner tree of minimum cost under a hop constraint which requires the depth of the tree to be at most $k$. Our main result is an exact algorithm for metrics induced by graphs with bounded treewidth that runs in time $n^{O(k)}$. For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if $k$ is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the $k$-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension and bounded doubling dimension. For non-metric graphs, we rule out an $o(\log n)$-approximation, assuming P$\neq$NP even when relaxing the hop constraint by any additive constant.

A hybrid multifactorial evolutionary algorithm and firefly algorithm for the clustered minimum routing cost tree problem

The clustered minimum routing cost tree (CluMRCT) problem is a recent problem with a wide range of real-life applications, especially in designing computer networks with peer-to-peer architecture. Many multifactorial evolutionary algorithms have been proposed to solve multiple CluMRCT problems simultaneously. However, these algorithms only function effectively on complete graphs with small-and-medium sizes. Moreover, the blindness and randomness in the transfer of genetic materials cause a reduction in the exploitation ability and make these algorithms ineffective to solve low-similarity tasks. This paper proposes a hybrid multitasking algorithm named multifactorial firefly algorithm, which integrates the firefly algorithm’s strong exploitation ability to enhance the self-evolution of each task when facing low-similarity tasks while improving inter-task knowledge transfers by delivering higher-quality solutions. Also, the proposed algorithm is equipped with new encoding and decoding to focus more on potential search areas on both complete and sparse graphs. The experiments and Wilcoxon signed-rank tests were conducted on various instances to verify our proposal with several state-of-the-art methods. The results portrayed that the proposed encoding scheme helped multitasking algorithms improve solution quality by 32% on average. Besides, the statistical test values proved the superiority of the proposed hybrid algorithm in terms of solution quality and convergence trend.

Efficient optimization of the Held–Karp lower bound

Given a weighted undirected graph G=(V,E), the Held–Karp lower bound for the Traveling Salesman Problem (TSP) is obtained by selecting an arbitrary vertex p ¯∈V, by computing a minimum cost tree spanning V∖{p ¯} and adding two minimum cost edges adjacent to p ¯. In general, different selections of vertex p ¯ provide different lower bounds. In this paper it is shown that the selection of vertex p ¯ can be optimized, to obtain the largest possible Held–Karp lower bound, with the same worst-case computational time complexity required to compute a single minimum spanning tree. Although motivated by the optimization of the Held–Karp lower bound for the TSP, the algorithm solves a more general problem, allowing for the efficient pre-computation of alternative minimum spanning trees in weighted graphs where any vertex can be deleted.

Estimating the Optimal Location for the Storage of Pellet Surplus

This paper deals with the problem of managing the surplus that arises during the seasonal production of pellets, which will be sold in the period of increased demand. Dijkstra’s algorithm is used in issues connected with finding a new storage place with a view of the optimisation of the transport costs of pellets produced by a company in 18 different towns in the Lubelskie Voivodeship in Poland. The most optimal location for the new pellet storage site has been determined, for which the total length of the traveled routes is the shortest, taking into account the actual shares of individual plants in the total production. The construction of the graph with the shortest paths was made on the basis of the existing network of available transport roads, and the nodes of the graph were their intersections. The most advantageous storage location of pellets was identified by the calculation the total transport cost using a minimum-cost tree of shortest paths. Based on the estimated transport assumptions, the lowest total cost of transport from all 18 plants was 3092.0 (km), which corresponds to an average distance to production plants of 89.7 km and 61.7 km to estimated selling distribution. The new storage point is suggested near the town of Piaski. Average cost of travel for all trees obtained for existing plant locations and subsequent distribution to points of sale was 4113.7 (km), while standard deviation 735.2 (km). Additionally, a relative increase in costs was estimated in the case of selecting other locations. Using spatial interpolation and geoprocessing tools, a map—showing the increase in pellet transport costs in relation to the most optimal solution—was developed. The constructed map allows for a better analysis of cost increases than a single point. It was stated that the increase in transport costs does not exceed 10% of lowest cost for 17.6% area of studied area. It was found that the most convenient area is shifted to the south of the voivodship and improperly adopted storage location can increase transport costs by up to 75%.

Heuristic Methods for Minimum-Cost Pipeline Network Design \u2013 a Node Valency Transfer Metaheuristic

Designing low-cost network layouts is an essential step in planning linked infrastructure. For the case of capacitated trees, such as oil or gas pipeline networks, the cost is usually a function of both pipeline diameter (i.e. ability to carry flow or transferred capacity) and pipeline length. Even for the case of incompressible, steady flow, minimizing cost becomes particularly difficult as network topology itself dictates local flow material balances, rendering the optimization space non-linear. The combinatorial nature of potential trees requires the use of graph optimization heuristics to achieve good solutions in reasonable time. In this work we perform a comparison of known literature network optimization heuristics and metaheuristics for finding minimum-cost capacitated trees without Steiner nodes, and propose novel algorithms, including a metaheuristic based on transferring edges of high valency nodes. Our metaheuristic achieves performance above similar algorithms studied, especially for larger graphs, usually producing a significantly higher proportion of optimal solutions, while remaining in line with time-complexity of algorithms found in the literature. Data points for graph node positions and capacities are first randomly generated, and secondly obtained from the German emissions trading CO2 source registry. As political will for applications and storage for hard-to-abate industry CO2 emissions is growing, efficient network design methods become relevant for new large-scale CO2 pipeline networks.

Serial and parallel memetic algorithms for the bounded diameter minimum spanning tree problem

AbstractGiven a connected, weighted, undirected graph G = (V, E) and an integer D ≥ 2, the bounded diameter minimum spanning tree (BDMST) problem seeks a spanning tree of minimum cost, whose diameter is no greater than D. The problem is known to be NP‐hard, and finds application in various domains such as information retrieval, wireless sensor networks and distributed mutual exclusion. This article presents a two‐phase memetic algorithm for the BDMST problem that combines a specialized recombination operator (proposed in this work) with good heuristics in order to more effectively direct the exploration of the search space into regions containing better solutions. A parallel meta‐heuristic is also proposed and shown to obtain very good speedups – super‐linear, in several cases – vis‐à‐vis the serial memetic algorithm. To the best of the authors' knowledge, this is the first parallel meta‐heuristic proposed for the BDMST problem, and potentially paves the way for handling much larger problems than reported in the literature. Some observations and theorems are presented in order to provide the underlying framework for the proposed algorithms. Further, the proposed memetic algorithm and parallel meta‐heuristic are shown, in the course of several computational experiments, to in general obtain superior solution quality with much lesser computational effort in comparison to the best known meta‐heuristics in the literature over a wide range of benchmark instances.

Semi-Supervised Hyperspectral Image Classification Based on Label Propagation via Selected Path

Most graph-based semi-supervised classification methods do not perform well in hyperspectral image classification tasks due to their high complexity and other limitations. This paper proposes a label propagation semi-supervised classification algorithm which uses a few selected important paths to considerably reduce computational costs. It establishes that the most important paths for label propagation are minimum cost paths. It proves that minimum cost paths exist in minimum cost trees (MCT), and proposes a method based on a variant minimum spanning tree (MST) combined with priority queue to construct MCTs. The algorithm propagates labels from unlabeled nodes to labeled ones, a unique way different from any other studies where propagation is in the opposite direction, which brings about several clear advantages. These include that only one propagation path is required for each unlabeled node, improving both timing and memory performance. It also helps to solve a problem posed by sparse graphs where some image pixels cannot be classified, a situation which is especially problematic in large-scale image classification. The proposed method has the advantages of linear computational complexity, is independent of data dimension, has fewer parameters and is insensitive to the values of parameters. Moreover, it does not need large numbers of labelled pixels nor complex training processes. Experiments on hyperspectral images have shown that, compared with several existing algorithms, the proposed method achieves better performance in less time. The paper addresses some fundamental issues regarding propagating labels in graph based semi-supervised classifications. Due to the simplicity and the fast speed of the algorithm, it is also suitable to be integrated into both state-of-the-art and future hyperspectral image classification frameworks which have a label propagation stage.

An exact approach for the Minimum-Cost Bounded-Error Calibration Tree problem

The Minimum-Cost Bounded-Error Calibration Tree problem (MBCT) is a wireless network optimization problem that arises from the sensors’ need of periodical calibration. The MBCT takes into account two objectives. The first is to minimize the communication distance between the network sensors, while the second is to reduce the maximum post-calibration error in the network. In this paper, we propose a mathematical formulation for the MBCT. Furthermore, we solve the problem using two different implementations of an Augmented $$\epsilon $$-Constraint Method ($$aug\,\epsilon \text {-}CM$$) that incorporate the proposed formulation. We also employed the Pareto-fronts obtained by $$aug\,\epsilon \text {-}CM$$ to evaluate the Node-depth Phylogenetic-based Non-dominated Sorting Artificial Immune System (NPE-NSAIS), the most recent heuristic in the literature for MBCT. Computational experiments showed that $$aug\,\epsilon \text {-}CM$$ can solve MBCT instances up to 50 nodes. Furthermore, a statistical test demonstrated that the running times of one of the $$aug\,\epsilon \text {-}CM$$ implementations was significantly smaller than those of the other. Finally, we show that NPE-NSAIS solutions are very close to the Pareto-fronts given by $$aug\,\epsilon \text {-}CM$$, achieving good results on the evaluated metrics.

Mining Relationships among Multiple Entities in Biological Networks.

Identifying topological relationships among multiple entities in biological networks is critical towards the understanding of the organizational principles of network functionality. Theoretically, this problem can be solved using minimum Steiner tree (MSTT) algorithms. However, due to large network size, it remains to be computationally challenging, and the predictive value of multi-entity topological relationships is still unclear. We present a novel solution called Cluster-based Steiner Tree Miner (CST-Miner) to instantly identify multi-entity topological relationships in biological networks. Given a list of user-specific entities, CST-Miner decomposes a biological network into nested cluster-based subgraphs, on which multiple minimum Steiner trees are identified. By merging all of them into a minimum cost tree, the optimal topological relationships among all the user-specific entities are revealed. Experimental results showed that CST-Miner can finish in nearly log-linear time and the tree constructed by CST-Miner is close to the global minimum.

Designing Cost-Effective Reliable Networks From a Risk Analysis Perspective: A Case Study for a Hospital Campus

The unavailability of information and communication services due to network-related incidents may have a significant impact on large organizations. Network incidents can hence be viewed as a risk for organizations whose consequences are not accounted for by traditional network design problems. In this work, we address the problem of designing a reliable wired network from a risk analysis perspective. We propose a novel methodology for the quantitative assessment of the risk associated with network-related incidents in a hospital campus. We then define an optimization problem to find the topology that minimizes the network cost plus the expected loss over time attributable to the unavailability of corporate services to staff affected by network incidents. A case study illustrates our methodology and its benefits. Using available public information, we design the topology of a campus network for a large hospital where the cost of labor exceeds 200M€/year. The solution to our optimization problem is found through well-known genetic algorithms and provides a topology where network nodes with a higher impact on productivity exhibit higher reliability. As a consequence, the topology obtained reduces more than 95% (+392 000€) the expected annual lost profits when compared to common reduced-cost topologies such as the minimum-cost ring or the non-reliable minimum-cost tree, showing that investment in risk reduction pays off. Our contribution may be used by engineers to (re)design cost-effective reliable networks or by hospital managers to support decisions on updating present infrastructure based on risk reduction.

Algorithms for the power-p Steiner tree problem in the Euclidean plane

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case.

The Minimum Routing Cost Tree Problem

The minimum routing cost tree problem arises when we need to find the tree minimizing the minimum travel/communication cost, i.e., the tree which presents the minimal difference with the same cost computed on the whole network. This paper provides the state of the art of the problem and proposes a new heuristic based on the identification of a core of the network around which the solution can be built. The algorithm has been tested on literature instances of up to one thousand nodes. The results, compared with those of other heuristic algorithms, prove the competitiveness of the proposed one both in terms of the quality of the solution and computation time.

Distributed Topology Construction in ZigBee Wireless Networks

Topology control is one of the important techniques in wireless multi-hop networks to preserve connectivity and extend the network lifetime. This is more significant in ZigBee, since the address assignment scheme is tightly coupled with topology construction. For example, there can be orphan nodes that cannot receive the network address and isolated from the network due to predefined network configurations. In this paper, we propose a distributed topology construction algorithm that controls the association time of each node in order to solve the orphan node problem in ZigBee as well as construct an efficient routing tree topology. The main idea of the distributed topology construction algorithm is to construct primary backbone nodes by propagating the invitation packets and controlling the association time based on the link quality. Since the dynamically selected primary nodes are spread throughout the network, they can provide backbone to accept the association requests from the remaining secondary nodes which are majority in a network. In the performance evaluation, we show that the proposed topology construction algorithm effectively solves the orphan node problem regardless of network density as well as provides efficient tree routing cost comparable to the approximation algorithm for degree constrained minimum routing cost tree (DC-MRCT) problem.

Two new heuristics for the dominating tree problem

Dominating Tree Problem (DTP) aims to find a dominating tree (d T r e e) of minimum cost on a given connected, undirected and weighted graph in such a way that a vertex in the graph is either in d T r e e or adjacent to a vertex in d T r e e. A solution (d T r e e) to this problem can be used as routing backbone in wireless sensor network. Being a \(\mathcal {NP}\)-Hard problem, several problem-specific heuristics and metaheuristic techniques have been proposed. This paper presents two new heuristics for the DTP. First one is a new problem-specific heuristic that exploits the problem structure effectively, whereas the other is an artificial bee colony (ABC) algorithm. The proposed ABC for the DTP is different from the existing ABC algorithm for the DTP in the literature on its two main components: initial solution generation, and determining a neighboring solution. Computational results show on a set of standard benchmark instances that the proposed problem-specific heuristic and ABC algorithm for the DTP demonstrate the superiority over all existing problem-specific heuristics and metaheuristic techniques respectively in the literature.

A two-level solution approach for solving the generalized minimum spanning tree problem

In this paper, we are addressing the generalized minimum spanning tree problem, denoted by GMSTP, which is a variant of the classical minimum spanning tree (MST) problem. The main characteristic of this problem is the fact that the vertices of the graph are partitioned into a given number of clusters and we are looking for a minimum-cost tree spanning a subset of vertices which includes exactly one vertex from each cluster. We describe a two-level solution approach for solving the GMSTP obtained by decomposing the problem into two logical and natural smaller subproblems: an upper-level (global) subproblem and a lower-level (local) subproblem and solving them separately. The goal of the first subproblem is to determine (global) trees spanning the clusters using a genetic algorithm with a diploid representation of the individuals, while the goal of the second subproblem is to determine the best tree (w.r.t. cost minimization), for the above mentioned global trees, spanning exactly one vertex from each cluster. The second subproblem is solved optimally using dynamic programming. Extensive computational results are reported and discussed for an often used set of benchmark instances. The obtained results show an improvement in the quality of the achieved solutions, and demonstrate the efficiency of our approach compared to the existing methods from the literature.