Constrained Steiner Tree Research Articles

A Lagrangean-based decomposition approach for the link constrained Steiner tree problem

The link constrained Steiner tree problem is a variant of the classic Steiner tree problem where the number of links to be activated must not exceed a pre-fixed value. We introduce a multi-start heuristic to obtain a quick feasible solution. The proposed heuristic is embedded into a decomposition framework based on Lagrangean relaxation. In particular, the relaxed problem is decomposed into two polynomially solvable subproblems and, to tackle the Lagrangean dual, we introduce a dual ascent procedure where just one multiplier at a time is updated. Our approach can be classified as a Lagrangean heuristic. In fact, at each iteration of the dual ascent procedure, the information derived from the solution of the relaxed problem is used to provide a feasible solution, by solving a restricted problem defined on an appropriate subgraph. Several versions of the proposed approach are defined and tested on instances drawn from the scientific literature.

Finding Outbreak Trees in Networks with Limited Information

Real-time control of infectious disease outbreaks represents one of the greatest epidemiological challenges currently faced. In this paper we address the problem of identifying contagion patterns responsible for the spread of a disease in a network, which can be applied in real-time to evaluate an ongoing outbreak. We focus on the scenario where limited information, i.e. infection reports which may or may not include the actual source, is available during an ongoing outbreak and we seek the most likely infection tree that spans at least a set of known infected nodes. This problem can be represented using a maximum likelihood constrained Steiner tree model where the objective is to find a spanning tree with an assignment of integer nodes weights. We propose a novel formulation and solution method based on a two-step heuristic which (1) reduces the initial graph using a polynomial time algorithm designed to find feasible infection paths and (2) solves an exact mixed integer linear programming reformulation of the maximum likelihood model on the resulting subgraph. The proposed methodology can be applied to outbreaks which may evolve from multiple sources. Simulated contagion episodes are used to evaluate the performance of our solution method. Our results show that the approach is computationally efficient and is able to reconstruct a significant proportion of the outbreak, even in the context of low levels of information availability.

The two-level diameter constrained spanning tree problem

In this article, we introduce the two-level diameter constrained spanning tree problem (2-DMSTP), which generalizes the classical DMSTP by considering two sets of nodes with different latency requirements. We first observe that any feasible solution to the 2-DMSTP can be viewed as a DMST that contains a diameter constrained Steiner tree. This observation allows us to prove graph theoretical properties related to the centers of each tree which are then exploited to develop mixed integer programming formulations, valid inequalities, and symmetry breaking constraints. In particular, we propose a novel modeling approach based on a three-dimensional layered graph. In an extensive computational study we show that a branch-and-cut algorithm based on the latter model is highly effective in practice.

Hop constrained Steiner trees with multiple root nodes

We consider a network design problem that generalizes the hop and diameter constrained Steiner tree problem as follows: Given an edge-weighted undirected graph with two disjoint subsets representing roots and terminals, find a minimum-weight subtree that spans all the roots and terminals so that the number of hops between each relevant node and an arbitrary root does not exceed a given hop limit H. The set of relevant nodes may be equal to the set of terminals, or to the union of terminals and root nodes. This article proposes integer linear programming models utilizing one layered graph for each root node. Different possibilities to relate solutions on each of the layered graphs as well as additional strengthening inequalities are then discussed. Furthermore, theoretical comparisons between these models and to previously proposed flow- and path-based formulations are given. To solve the problem to optimality, we implement branch-and-cut algorithms for the layered graph formulations. Our computational study shows their clear advantages over previously existing approaches.

Niched ant colony optimization with colony guides for QoS multicast routing

Quality-of-service (QoS) multicast routing is essential to many network applications such as IPTV, Internet radio, multimedia broadcasting, and real-time telecommunication. Recently, the minimum-cost multicast tree with the delay-and-bandwidth constraint (MinC/DB) problem in QoS multicast routing has caused the most attention. As the MinC/DB problem can be reduced to the constrained Steiner tree problem which has been shown to be NP-complete, it is very unlikely that a polynomial time algorithm would exist. In this paper, we propose a niched ant colony optimization with colony guides (NACOg) algorithm to tackle the MinC/DB problem. The NACOg algorithm first deliberates a constrained tree traversal (CTT) strategy that guarantees the search to any feasible trees with respect to the QoS constraints. The proposed CTT strategy employs adaptive memory structure as contemplated in the tabu search and the strategy is more effective in constraint-handling than both the penalty-function and the produce-and-repair strategies which were broadly used in the literature. The evolutionary optimization of the NACOg algorithm is empowered by the search balance between diversification and intensification. The diversification search is practiced by the individual niche-colony with its own pheromone matrix, and the intensification search is facilitated by the learning scheme which refers to the experience obtained by the niche-colony guide and the entire-colony guide. The experimental results on 100 problem instances have shown that the proposed NACOg algorithm produces the least cost QoS multicast trees compared to those obtained by the Haghighat genetic algorithm and the well-known KPP heuristic.

A hybrid scatter search meta-heuristic for delay-constrained multicast routing problems

This paper investigates the first hybrid scatter search and path relinking meta-heuristic for the Delay-Constrained Least-Cost (DCLC) multicast routing problem. The underpinning mathematic model of the DCLC multicast routing problem is the constrained Steiner tree problem in graphs, a well known NP-complete problem. After combining a path relinking method as the solution combination method in scatter search, we further explore two improvement strategies: tabu search and variable neighborhood search, to intensify the search in the hybrid scatter search algorithm. A large number of simulations on some benchmark instances from the OR-library and a group of random graphs of different characteristics demonstrate that the improvement strategy greatly affects the performance of the proposed scatter search algorithm. The hybrid scatter search algorithm intensified by a variable neighborhood descent search is highly efficient in solving the DCLC multicast routing problem in comparison with other algorithms and heuristics in the literature.

Multicast routing for delay variation bound using a modified ant colony algorithm

The delay and delay variation-bounded Steiner tree problem is an important multicast routing problem in real-time multimedia networks. Such a constrained Steiner tree problem is known to be NP-complete. This paper proposes an ant colony algorithm with orientation factor and applies it to multicast routing problem with the constraints of delay variation bound. The orientation factor enables the ant to get rid of the initial blindness when searching paths, makes use of the search results and reduces the misguiding effect of pheromone on irrelevant paths, thus overcoming the drawbacks of slow convergence existing in the basic ant colony algorithm, increasing the speed of convergence and speeding up the finding of feasible solution to the problem. The simulation results show that the modified algorithm makes it possible to find a feasible solution to the multicast routing problem with delay variation bound. Compared with the conventional ant colony algorithm, the convergence speed of the modified algorithm is improved.

Ant-based distributed constrained steiner tree algorithm for jointly conserving energy and bounding delay in ad hoc multicast routing

The minimum-energy multicast tree problem aims to construct a multicast tree rooted at the source node and spanning all the destination nodes such that the sum of transmission power at non-leaf nodes is minimized. However, aggressive power assignment at non-leaf nodes, although conserving more energy, results in multicast trees that suffer from higher hop count and jeopardizes delay-sensitive applications, signifying a clear tradeoff between energy efficiency and delay. This article formulates these issues as a constrained Steiner tree problem, and describes a distributed constrained Steiner tree algorithm, which jointly conserves energy and bounds delay for multicast routing in ad hoc networks. In particular, the proposed algorithm concurrently constructs a constrained Steiner tree, performs transmission power assignment at non-leaf nodes, and strives to minimize the sum of transmission power of non-leaf nodes, subject to the given maximum hop count constraint. Simulation results validate the effectiveness and reveal the characteristics of the proposed algorithm.

Dynamic multicast routing algorithm for delay and delay variation-bounded Steiner tree problem

The delay and delay variation-bounded Steiner tree problem is an important problem in real-time multimedia networks, and is known to be NP-complete. In this paper, we propose an efficient heuristic multicast routing algorithm based on simulated annealing named SADDVMA to construct the constrained Steiner tree. To avoid enlargement of search area and increase of computing time, the proposed heuristic algorithm uses a procedure called Paths-switching to construct neighbors in feasible region according to the relationship between delay and delay variation. We also give a method to dynamically reorganize the multicast tree in response to changes for the destinations. Simulations demonstrate that our algorithm is better in terms of tree cost as compared to the existing algorithms. Further, it performs excellent performance of delay and delay variation, rapid convergence and better real-time property.

Distributed multicast routing for delay and delay variation-bounded Steiner tree using simulated annealing

The delay and delay variation-bounded Steiner tree problem is an important multicast routing issue in real-time multimedia networks. Such a constrained Steiner tree problem is known to be NP-complete. In this paper, we propose a distributed multicast routing algorithm based on simulated annealing to produce routing trees having a minimal network cost under delay and delay variation constraints. The proposed algorithm is fully distributed, and supports the dynamic reorganizing of the multicast tree in response to changes for the destination. Simulations demonstrate that our algorithm is better in terms of tree cost as compared with other existing algorithms, and it performs excellently in delay and delay variation. Furthermore, it has high success ratio, rapid convergence and better real-time property.

Constrained Steiner trees in Halin graphs

In this paper, we study the problem of computing a minimum cost Steiner tree subject to a weight constraint in a Halin graph where each edge has a nonnegative integer cost and a nonnegative integer weight. We prove the NP-hardness of this problem and present a fully polynomial time approximation scheme for this NP-hard problem.

GA-based heuristic algorithms for bandwidth-delay-constrained least-cost multicast routing

The bandwidth-delay-constrained least-cost multicast routing is a challenging problem in high-speed multimedia networks. Computing such a constrained Steiner tree is an NP-complete problem. In this paper, we propose several novel solutions to this problem based on genetic algorithms (GA). The proposed solutions consist of six different schemes for genotype representation, and also several new heuristic algorithms for mutation, crossover, and creation of random individuals. We evaluate the performance and efficiency of the proposed algorithms in comparison with other existing heuristic and GA-based algorithms using simulation results. The most efficient combination of various proposed alternative algorithms is selected as our final solution by the simulation results. This proposed GA-based algorithm has overcome the existing algorithms considering average tree cost and running time.

A PTAS for weight constrained Steiner trees in series–parallel graphs

In this paper, we study the problem of computing a minimum cost Steiner tree subject to weight constraint in a series–parallel graph where each edge has a nonnegative integer cost and a nonnegative integer weight. We present a strongly polynomial time approximation scheme for this NP-complete problem.

Algorithms for delay-constrained low-cost multicast tree construction

With the proliferation of multimedia group applications, the construction of multicast trees satisfying quality of service (QoS) requirements is becoming a problem of prime importance. Multicast groups are usually classified as sparse or pervasive groups depending on the physical distribution of group members. They are also classified based on the temporal characteristics of group membership into static and dynamic groups. In this paper, we propose two algorithms for constructing multicast trees for multimedia group communication in which the members are sparse and static. The proposed algorithms use a constrained distributed unicast routing algorithm for generating low-cost, bandwidth and delay constrained multicast trees. These algorithms have lower message complexity and call setup time due to their nature of iteratively adding paths, rather than edges, to partially constructed trees. We study the performance (in terms of call acceptance rate, call setup time and multicast tree cost) of these algorithms through simulation by comparing them with that of a recently proposed algorithm (V. Kompella, J.C. Pasquale, G.C. Polyzos, Two distributed algorithms for the constrained Steiner tree problem, in: Proc. Comp. Comm. Networking, San Diego, CA, June 1993) for the same problem. The simulation results indicate that the proposed algorithms provide larger call acceptance rates, lower setup times and comparable tree costs.

Neural and delay based heuristics for the Steiner problem in networks

High speed networks such as the B-ISDN must be adequately equipped to handle multipoint communication in a fast and economical manner. Multicast applications include desktop video conferencing, distance learning, distributed database applications, etc. In networks employing the asynchronous transfer mode (ATM) technology, routing a multicast is achieved by constructing a tree that spans the source and all the destinations. For the purpose of routing, the network is modeled as a weighted, undirected graph. The graph-theoretic solution is to find a minimum Steiner tree for the graph given a set of destinations. This formulation suffices for building multicast trees with a single optimization constraint as would be the xcase for best effort transport. For real-time traffic, however, it is necessary to ensure that the delay between the sender and each of the receivers is bounded. In this case the network is modeled as an undirected graph, where the edges have both a cost and a delay associated with them. The graph-theoretic solution is then to find a constrained minimum Steiner tree such that the delay between the source and each of the destinations does not violate the specified bound. Both of these problems are NP-complete. In this paper we review prior work on the multipoint routing problem and discuss the formulation of the unconstrained and constrained Steiner problems. We use the random neural network (RNN) to significantly improve the quality of trees found by the two existing best heuristics for finding Steiner trees - the minimum spanning tree heuristic and the average distance heuristic. We also develop a new heuristic for finding delay constrained Steiner trees. Experimental results are presented which show that the new heuristics improve significantly over existing ones.

Evaluation of multicast routing algorithms for real-time communication on high-speed networks

Multicast (MC) routing algorithms capable of satisfying the quality of service (QoS) requirements of real-time applications will be essential for future high-speed networks. We compare the performance of all of the important MC routing algorithms when applied to networks with asymmetric link loads. Each algorithm is judged based on the quality of the MC trees it generates and its efficiency in managing the network resources. Simulation results over random networks show that unconstrained algorithms are not capable of fulfilling the QoS requirements of real-time applications in wide-area networks. Simulations also reveal that one of the unconstrained algorithms, reverse path multicasting (RPM), is quite inefficient when applied to asymmetric networks. We study how combining routing with resource reservation and admission control improves the RPM's efficiency in managing the network resources. The performance of one semiconstrained heuristic, MSC, three constrained Steiner tree (CST) heuristics, Kompella, Pasquale, and Polyzos (1992), constrained adaptive ordering (CAO), and bounded shortest multicast algorithm (BSMA), and one constrained shortest path tree (CSPT) heuristic, the constrained Dijkstra heuristic (CDKS) are also studied. Simulations show that the semiconstrained and constrained heuristics are capable of successfully constructing MC trees which satisfy the QoS requirements of real-time traffic. However, the cost performance of the heuristics varies. The BSMA's MC trees are lower in cost than all other constrained heuristics. Finally, we compare the execution times of all algorithms, unconstrained, semiconstrained, and constrained.

A constrained Steiner tree problem

The Steiner tree problem on a graph involves finding a minimum cost tree which connects a designated subset of the nodes in the graph. Variants of the basic Steiner tree model can arise in the design of telecommunication networks where customers must be connected to a switching center. In this paper, we consider the constrained Steiner tree problem which is the Steiner tree problem on graph with one additional side constraint that imposes a budget on the total amount of a resource (e.g., employee maintenance time) required by the arcs in the solution. We discuss various problem formulations, decomposition based solution approaches, and computational experience with the proposed methods.