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Greedy Heuristics and an Evolutionary Algorithm for the BoundedDiameter Minimum Spanning Tree Problem
 Proceedings of the 2003 ACM Symposium on Applied Computing
, 2003
"... bound D, the boundeddiameter minimum spanning tree problem seeks a spanning tree on G of lowest weight in which no path between two vertices contains more than D edges. This problem is NPhard for 4 1, where n is the number of vertices in G. An existing greedy heuristic for the problem, called ..."
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Cited by 37 (13 self)
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bound D, the boundeddiameter minimum spanning tree problem seeks a spanning tree on G of lowest weight in which no path between two vertices contains more than D edges. This problem is NPhard for 4 1, where n is the number of vertices in G. An existing greedy heuristic for the problem, called OTTC, is based on Prim's algorithm. OTTC usually yields poor results on instances in which the triangle inequality approximately holds; it always uses the lowestweight edges that it can, but such edges do not in general connect the interior nodes of lowweight boundeddiameter trees. A new randomized greedy heuristic builds a boundeddiameter spanning tree from its center vertex or vertices. It chooses each next vertex at random but attaches the vertex with the lowestweight eligible edge. This algorithm is faster than OTTC and yields substantially better solutions on Euclidean instances. An evolutionary algorithm encodes spanning trees as lists of their edges, augmented with their center vertices. It applies operators that maintain the diameter bound and always generate valid o#spring trees. These operators are e#cient, so the algorithm scales well to larger problem instances. On 25 Euclidean instances of up to 1 000 vertices, the EA improved substantially on solutions found by the randomized greedy heuristic.
A PermutationCoded Evolutionary Algorithm for the BoundedDiameter Minimum Spanning Tree Problem
 in 2003 Genetic and Evolutionary Computation Conference’s Workshops Proceedings, Workshop on Analysis and Desgn of Representations
, 2003
"... The diameter of a tree is the largest number of edges on any path between two vertices in it. Given a weighted, connected, undirected graph G and a bound D 2, the boundeddiameter minimum spanning tree problem seeks a spanning tree on G of minimum weight whose diameter does not exceed D. ..."
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Cited by 11 (3 self)
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The diameter of a tree is the largest number of edges on any path between two vertices in it. Given a weighted, connected, undirected graph G and a bound D 2, the boundeddiameter minimum spanning tree problem seeks a spanning tree on G of minimum weight whose diameter does not exceed D. An evolutionary algorithm for this NPhard problem encodes candidate trees as permutations of their vertices. The first vertex (if D is even) or the first two vertices (if D is odd) form the center of the tree a permutation represents. A greedy heuristic appends the remaining vertices to the tree in their listed order, as economically as possible, while maintaining the diameter bound. In tests on 25 Euclidean problem instances, this EA identifies shorter trees on average than does an EA that encodes trees as sets of their edges, though it takes longer.
Combining a Memetic Algorithm with Integer Programming to Solve the PrizeCollecting Steiner Tree Problem
 Proceedings of the Genetic and Evolutionary Computation Conference (GECCO2004), volume 3102 of LNCS
, 2004
"... The prizecollecting Steiner tree problem on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. For this wellknown problem we develop a new algorithmic ..."
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Cited by 11 (4 self)
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The prizecollecting Steiner tree problem on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. For this wellknown problem we develop a new algorithmic framework consisting of three main parts: (1) An extensive preprocessing phase reduces the given graph without changing the structure of the optimal solution. (2) The central part of our approach is a memetic algorithm (MA) based on a steadystate evolutionary algorithm and an exact subroutine for the problem on trees. (3) The solution population of the memetic algorithm provides an excellent starting point for postoptimization by solving a relaxation of an integer linear programming (ILP) model constructed from a model for finding the minimum Steiner arborescence in a directed graph.
Initialization is Robust in Evolutionary Algorithms that Encode Spanning Trees as Sets of Edges
 in Proceedings of the 2002 ACM Symposium on Applied Computing
, 2002
"... Evolutionary algorithms (EAs) that search spaces of spanning trees can encode candidate trees as sets of edges. In this case, edgesets for an EA's initial population should represent spanning trees chosen with uniform probabilities on the graph that underlies the target problem instance. The g ..."
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Cited by 7 (1 self)
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Evolutionary algorithms (EAs) that search spaces of spanning trees can encode candidate trees as sets of edges. In this case, edgesets for an EA's initial population should represent spanning trees chosen with uniform probabilities on the graph that underlies the target problem instance. The generation of random spanning trees is not as simple as it might appear. Mechanisms based on Prim's and Kruskal's minimum spanning tree algorithms are not uniform, and uniform mechanisms are slow, not guaranteed to terminate, or require that the underlying graph be complete.
Designing Reliable Communication Networks with a Genetic Algorithm Using a Repair Heuristic
, 2003
"... This paper investigates GA approaches for solving the reliable communication network design problem. For solving this problem a graph with minimum cost must be found that satisfies a given network reliability constraint. To consider the additional reliability constraint different approaches are poss ..."
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Cited by 5 (1 self)
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This paper investigates GA approaches for solving the reliable communication network design problem. For solving this problem a graph with minimum cost must be found that satisfies a given network reliability constraint. To consider the additional reliability constraint different approaches are possible. We show that existing approaches using penalty functions can result in invalid solutions and are therefore not appropriate for solving this problem. To overcome these problems we present a repair heuristic, which is based on the number of spanning trees in a graph. This heuristic always generates a valid solution, which when compared to a greedy cheapest repair heuristic shows that the new approach finds better solutions with less computational effort.
On the Optimal Communication Spanning Tree Problem
, 2003
"... This paper presents an investigation into the properties of the optimal communication spanning tree (OCST) problem. The OCST problem finds a spanning tree that connects all nodes and satisfies their communication requirements for a minimum total cost. The paper compares the properties of randomly ..."
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Cited by 5 (4 self)
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This paper presents an investigation into the properties of the optimal communication spanning tree (OCST) problem. The OCST problem finds a spanning tree that connects all nodes and satisfies their communication requirements for a minimum total cost. The paper compares the properties of randomly created solutions to the best solutions that are found using an evolutionary algorithm framework. The results show that on average the distance between the optimal solution and the minimum spanning tree (MST) that is calculated according to the distance weights is significantly smaller than the distance between a randomly created solution and the MST. This means, optimal solutions for the OCST problem are biased towards the MST defined on the distance weights alone. Consequently, the performance of optimization methods for the OCST problem can be increased if the search is biased towards MSTlike solutions.
Codings and operators in two genetic algorithms for the leafconstrained minimum spanning tree problem
 International Journal of Applied Mathematics and Computer Science
, 2004
"... The features of an evolutionary algorithm that most determine its performance are the coding by which its chromosomes represent candidate solutions to its target problem and the operators that act on that coding. Also, when a problem involves constraints, a coding that represents only valid solution ..."
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Cited by 4 (1 self)
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The features of an evolutionary algorithm that most determine its performance are the coding by which its chromosomes represent candidate solutions to its target problem and the operators that act on that coding. Also, when a problem involves constraints, a coding that represents only valid solutions and operators that preserve that validity represent a smaller search space and result in a more effective search. Two genetic algorithms for the leafconstrained minimum spanning tree problem illustrate these observations. Given a connected, weighted, undirected graph G with n vertices and a bound ℓ, this problem seeks a spanning tree on G with at least ℓ leaves and minimum weight among all such trees. A greedy heuristic for the problem begins with an unconstrained minimum spanning tree on G, then economically turns interior vertices into leaves until their number reaches ℓ. One genetic algorithm encodes candidate trees with Prüfer strings decoded via the Blob Code. The second GA uses strings of length n−ℓ that specify trees ’ interior vertices. Both GAs apply operators that generate only valid chromosomes. The latter represents and searches a much smaller space. In tests on 65 instances of the problem, both Euclidean and with weights chosen randomly, the BlobCoded GA cannot compete with the greedy heuristic, but the subsetcoded GA consistently identifies leafconstrained spanning trees of lower weight than the greedy heuristic does, particularly on the random instances.
Biased mutation operators for subgraphselection problems
 IN IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 2004
"... Many graph problems seek subgraphs of minimum weight that satisfy a set of constraints. Examples include the minimum spanning tree problem (MSTP), the degreeconstrained minimum spanning tree problem (dMSTP), and the traveling salesman problem (TSP). Lowweight edges predominate in optimum solution ..."
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Cited by 4 (0 self)
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Many graph problems seek subgraphs of minimum weight that satisfy a set of constraints. Examples include the minimum spanning tree problem (MSTP), the degreeconstrained minimum spanning tree problem (dMSTP), and the traveling salesman problem (TSP). Lowweight edges predominate in optimum solutions to such problems, and the performance of evolutionary algorithms (EAs) is often improved by biasing variation operators to favor these edges. We investigate the impact of biased edgeexchange mutation. In a largescale empirical investigation, we study the distributions of edges in optimum solutions of the MSTP, the dMSTP, and the TSP in terms of the edges ’ weightbased ranks. We approximate these distributions by exponential functions and derive approximately optimal probabilities for selecting edges to be incorporated into candidate solutions during mutation. A theoretical analysis of the expected running time
An evolutionary algorithm with solution archive for the generalized minimum spanning tree problem
 Proceedings of the 13th International Conference on Computer Aided Systems Theory: Part I, volume 6927 of LNCS
, 2012
"... We consider the recently proposed concept of enhancing an evolutionary algorithm (EA) with a complete solution archive. It stores evaluated solutions during the optimization in order to detect duplicates and to efficiently transform them into yet unconsidered solutions. For this approach we introdu ..."
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Cited by 2 (1 self)
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We consider the recently proposed concept of enhancing an evolutionary algorithm (EA) with a complete solution archive. It stores evaluated solutions during the optimization in order to detect duplicates and to efficiently transform them into yet unconsidered solutions. For this approach we introduce the socalled bounding extension in order to identify and prune branches in the triebased archive which only contain inferior solutions. This extension enables the EA to concentrate the search on promising areas of the solution space. Similarly to the classical branchandbound technique, bounds are obtained via primal and dual heuristics. As an application we consider the generalized minimum spanning tree problem where we are given a graph with nodes partitioned into clusters and exactly one node from each cluster must be connected in the cheapest way. As the EA uses operators based on two dual representations, we exploit two corresponding tries that complement each other. Test results on TSPlib instances document the strength of this concept and that it can compete with the leading metaheuristics for this problem in the literature.
Genetic Algorithm for the Transportation Problem with Discontinuous Piecewise Linear Cost Function
, 2006
"... We investigate a transportation problem with discontinuous piecewise linear cost function in this paper. The basic feasible solution for the transportation problem, in which flow bounds on edges are uncertain, is obtained by disaggregating piecewise linear cost function. A genetic algorithm is propo ..."
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We investigate a transportation problem with discontinuous piecewise linear cost function in this paper. The basic feasible solution for the transportation problem, in which flow bounds on edges are uncertain, is obtained by disaggregating piecewise linear cost function. A genetic algorithm is proposed to solve the transportation problem based on the network representation of the basic feasible solution. A basic feasible solution is represented by the spanning tree that consists of all basic edges and nonbasic edges with positive flow amount. Sorted edge set is used to represent the spanning tree. Edges of a spanning tree are sorted by root first search pattern in the representation. Single point crossover operator and random pivot mutation operator are developed for effective and efficient evolution. Computational tests demonstrate our genetic algorithm not only can obtain better solution quality, but also is more efficient than the previous proposed genetic algorithm based on the matrix encoding. Key words: Transportation problem; discontinuous piecewise linear cost function; genetic algorithm; spanning tree 1.