Solving the Graphical Steiner Tree Problem Using Genetic Algorithms

Steiner tree problem (STP) in graphs aims to find a tree of minimum weight in the graph that connects a given set of vertices. It is a classic NP-hard combinatorial optimization problem and has many applications in real world. Many approximate algorithms have been developed for STP, but they suffer from high computational complexity and weak worst-case solution guarantees, respectively. Heuristic algorithms are also developed. But each of them requires application domain knowledge to design and is only suitable for specific scenarios. In this paper, we design a novel framework Cherrypick based on novel graph embedding and deep reinforcement learning to tackle the STP. Given an STP instance, Cherrypick uses this embedding to encode its path-related information and sends the encoded graph to a deep reinforcement learning component based on deep Q network (DQN) to find solutions. We implement the Cherrypick and demonstrate its efficacy and efficiency with extensive experiments using real-world and synthetic datasets.

The Steiner tree problem in graphs (SPG) is one of the most studied problems in combinatorial optimization because of its theories and applications. It is one of the foundations to develop a Wireless Sensor Network (WSNs), such as multicast and topology design. SPG is an NP-Hard problem, and many heuristic and approximation algorithms have been proposed. Thus, this study proposes a Hybrid Genetic algorithm (HGA) to solve SPG. This study is the binary string representation for a set of chosen edges. To increase the diversity of the population and avoid falling into local optimization, we use a 2-longest Distance strategy, dynamic crossover rate, and chosen solutions must differ by at least 5%. The experiment results show that the HGA algorithm's running time equals 153.83% of the GA algorithm's, the deviation found by HGA, and the optimal distance only equals 65% that of GA.

In this paper we present the SteinLib, a library of data sets for the Steiner tree problem in graphs. This library extends former libraries on Steiner tree problems by many new interesting and difficult instances, most of them arising from real-world applications. We give a survey on the difficulty of these problem instances by stating references to state-of-the-art software packages that were the first or are currently among the best to solve these instances.

This paper presents an algorithm to obtain near optimal solutions for the Steiner tree problem in graphs. It is based on a Lagrangian relaxation of a multi-commodity flow formulation of the problem. An extension of the subgradient algorithm, the volume algorithm, has been used to obtain lower bounds and to estimate primal solutions. It was possible to solve several difficult instances from the literature to proven optimality without branching. Computational results are reported for problems drawn from the SteinLib library.

The class Steiner minimal tree problem is an extension of the standard Steiner minimal tree problem in graphs, motivated by the problem of wire routing in the area of physical design of very large scale integration (VLSI). This problem is NP-hard, even if there are no Steiner nodes and the graph is a tree; moreover, there exists no polynomial time approximation algorithm with a constant bound on the relative error under the hypothesis that P \(\neq\) NP [16]. Hence, fast and good heuristic algorithms are needed in practice. In this paper, we present an integer programming formulation of the problem. Using Lagrangean relaxation and subgradient optimization, we derive a lower bound. In order to test the lower bound, we present a procedure for generating test problems for the class Steiner minimal tree problem that have known optimal solutions. The computational experiments for the test problems demonstrate that the lower bound is very tight and differs from the optimal solutions by only a few percent on average for sparse graphs.

In the Steiner Tree Problem in Graphs, a subset of nodes, called terminals, must be efficiently interconnected. The graph is undirected and weighted, and candidate solutions can include additional nodes called Steiner vertices. The problem is NP-complete, and optimization algorithms based on population metaheuristics, e.g., Genetic Algorithms (GAs), have been proposed. However, traditional recombination operators may produce inefficient solutions for graph-based problems. We propose a new recombination operator for the Steiner Tree Problem based on the graph representation. The new operator is a kind of partition crossover: it breaks the graph formed by the union of two parents solutions and decomposes the evaluation function by finding connected subgraphs. We also investigate two soft-repair operators that produce small changes in candidate solutions: an MST transformation and a pruning repair. The GA with the proposed crossover operator was able to find the global optimum solution for all tested instances and has a success rate of 28\% in the worst case. The experiments with the proposed crossover presented a quick convergence to optimal solutions, and an average solution cost lower in most cases than other approaches.

Connecting a given set of points at minimum cost may be rated as one of the most important problems in telecommunications network design. Related questions may be formulated in metric spaces as well as in graphs. Given a weighted graph, the Steiner tree problem in graphs asks to determine a minimum cost subgraph spanning a set of specified vertices. This problem may be viewed as the combinatorial optimization problem in telecommunications. In this chapter, we survey Steiner problems from a telecommunications perspective with a special emphasis on the problem in graphs.

We consider the problem of finding a shortest rectilinear Steiner tree for a given set of points in the plane in the presence of rectilinear obstacles. The Steiner tree is allowed to run over obstacles; however, if we intersect the Steiner tree with some obstacle, then no connected component of the induced subtree must be longer than a given fixed length L. This kind of length restriction is motivated by its application in VLSI design where a large Steiner tree requires the insertion of buffers (or inverters) which must not be placed on top of obstacles. We show that the length-restricted Steiner tree problem can be approximated with a performance guarantee of 2 in O(n logn) time, where n denotes the size of the associated Hanan grid. Optimal length-restricted Steiner trees can be characterized to have a special structure. In particular, we prove that a certain graph, which is a variant of the Hanan grid, always contains an optimal solution. Based on this structural result, we can improve the performance guarantee of approximation algorithms for the special case that all obstacles are of rectangular shape or of constant complexity, i.e. they are represented by at most a constant number of edges. For such a scenario, we give a $\frac{5}{4}\alpha$ -approximation and a $\frac{2k}{2k-1}\alpha$ -approximation for any integral k ≥ 4, where α denotes the performance guarantee for the ordinary Steiner tree problem in graphs.

Given a graph G=(V,E) with node weight w:V→R+ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(nO(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+e)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+e)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.

The Steiner tree problem asks for a shortest subgraph connecting a given set of terminals in a graph. It is known to be APX-complete, which means that no polynomial time approximation scheme can exist for this problem, unless P=NP. Currently, the best approximation algorithm for the Steiner tree problem has a performance ratio of 1.55, whereas the corresponding lower bound is smaller than 1.01. In this paper, we provide for several Steiner tree approximation algorithms lower bounds on their performance ratio that are much larger. For two algorithms that solve the Steiner tree problem on quasi-bipartite instances, we even prove lower bounds that match the upper bounds. Quasi-bipartite instances are of special interest, as currently all known lower bound reductions for the Steiner tree problem in graphs produce such instances.

The increasing development of real-time multimedia network applications, many of which require multiple participants, has created the need for efficient multicast routing algorithms. Examples of such applications include video and tele-conferencing, video-on-demand, tele-medicine, distance education, etc. Several of them require multicasting with a certain Quality of Service (QoS) with respect to elements such as delay or bandwidth. This paper deals with Delay-Constrained Multicast Routing (DCMR) where the maximum end-to-end delay in a multicast session is bounded. The DCMR problem can be reduced to the Constrained Minimum Steiner Tree Problem in Graphs (CMStTG) which has been proven to be NP-complete. As a result, several heuristics have been developed to help solve it. In this paper, we developed a GRASP heuristic for the DCMR problem. Computational experiments on medium sized problems (50-100 nodes) from literature and comparison with existing algorithms have shown that the suggested GRASP heuristic is superior in quality for this set of problems.

The minimum connected dominating set (MCDS) problem is an important extension of the minimum dominating set problem, with wide applications, especially in wireless networks. Most previous works focused on solving MCDS problem in graphs with relatively small size, mainly due to the complexity of maintaining connectivity. This paper explores techniques for solving MCDS problem in massive real-world graphs with wide practical importance. Firstly, we propose a local greedy construction method with reasoning rule called 1hopReason. Secondly and most importantly, a hybrid dynamic connectivity maintenance method (HDC+) is designed to switch alternately between a novel fast connectivity maintenance method based on spanning tree and its previous counterpart. Thirdly, we adopt a two-level vertex selection heuristic with a newly proposed scoring function called chronosafety to make the algorithm more considerate when selecting vertices. We design a new local search algorithm called FastCDS based on the three ideas. Experiments show that FastCDS significantly outperforms five state-of-the-art MCDS algorithms on both massive graphs and classic benchmarks.

Metaheuristic algorithms for computing capacitated dominating set with uniform and variable capacities

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

In this paper, we propose a method of genetic algorithm (GA) for text clustering based on singular value decomposition technique. The main difficulty in the application of GA to text clustering is its long string representation in high dimensional space. Because the most straightforward and popular approach represents texts with vector space model (VSM), that is, each unique term in the vocabulary represents one dimension. Singular value decomposition (SVD) is a successful technique arising from numerical linear algebra that is used in latent semantic indexing (LSI). Employing the SVD-based document representation, LSI can overcome the problems by using statistically derived conceptual indices instead of individual words and provide a dimension reduced space. Genetic algorithm belongs to search techniques which could automatically exploit the optimal solution for objective or fitness function of an optimization problem. GA can be used in conjunction with the reduced latent semantic structure and improve clustering efficiency and accuracy. Our algorithm is performed on Reuter documents collection. The results show that the performance of SVD-based GA is significantly superior to that of conventional GA in vector space model.