The fundamental design choices in an evolutionary algorithm are its representation of candidate solutions and the operators that will act on that representation. We propose representing spanning trees in evolutionary algorithms for network design problems directly as sets of their edges, and we describe initialization, recombination, and mutation operators for this representation. The operators offer locality, heritability, and computational efficiency. Initialization and recombination depend on an underlying random spanning tree algorithm; three choices for this algorithm, based on the minimum spanning tree algorithms of Prim and Kruskal and on random walks, respectively, are examined analytically and empirically. We demonstrate the usefulness of the edge-set encoding in an evolutionary algorithm for the NP-hard degree-constrained minimum spanning tree problem. The algorithm's operators are easily extended to generate only feasible spanning trees and to incorporate local, problem-specific heuristics. Comparisons of this algorithm to others that encode candidate spanning trees via the Blob Code, with network random keys, and as strings of weights indicate the superiority of the edge-set encoding, particularly on larger instances.

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Given a connected, weighted, undirected graph G and a bound D, the bounded diameter minimum spanning tree problem seeks a spanning tree on G of minimum weight among the trees in which no path between two vertices contains more than D edges. In Prim's algorithm, the diameter of the growing spanning tree can always be known, so it is a good starting point from which to develop greedy heuristics for the bounded diameter problem. Abdalla, Deo, and Gupta described such an algorithm. It imitates Prim's algorithm but avoids edges whose inclusion in the spanning tree would violate the diameter bound. Running the algorithm from one start vertex requires time that is O(n 3 ).A modification of this approach uses the start vertex as the center of the spanning tree (if D is even) or as one of the two center vertices (if D is odd). This yields a simpler algorithm whose time is O(n 2 ). A further modification chooses each next vertex at random rather than greedily, though it still connects each vertex to the growing tree with the lowest-weight feasible edge. On Euclidean problem instances with small diameter bounds, the randomized heuristic is superior to the two fully greedy algorithms, though its advantage fades as the diameter bound grows. On instances whose edge weights have been chosen at random, the fully greedy algorithms outperform the randomized heuristic. ACM Reference Format:Julstrom. B. A. 2009. Greedy heuristics for the bounded diameter minimum spanning tree problem. ACM

The coding by which chromosomes represent candidate solutions is a fundamental design choice in a genetic algorithm. This paper describes a novel coding of spanning trees in a genetic algorithm for the degree-constrained minimum spanning tree problem. For a connected, weighted graph, this problem seeks to identify the shortest spanning tree whose degree does not exceed an upper bound k > 2. In the coding, chromosomes are strings of numerical weights associated with the target graph's vertices. The weights temporarily bias the graph's edge costs, and an extension of Prim's algorithm, applied to the biased costs, identifies the feasible spanning tree a chromosome represents. This decoding algorithm enforces the degree constraint, so that all chromosomes represent valid solutions and there is no need to discard, repair, or penalize invalid chromosomes. On a set of hard graphs whose unconstrained minimum spanning trees are of high degree, a genetic algorithm that uses this coding identifies degree-constrained minimum spanning trees that are on average shorter than those found by several competing algorithms.

Among the many codings of spanning trees for evolutionary search are those based on bijections between Prüfer strings-strings of n−2 vertex labels-and spanning trees on the labeled vertices. One of these bijections, called the Blob Code, showed promise as an evolutionary coding, but EAs that use it to represent spanning trees have not performed well. Here, a genetic algorithm that represents spanning trees via the Blob Code is faster than, and returns results competitive with those of, a GA that encodes spanning trees as edge-sets on Euclidean instances of the minimum routing cost spanning tree problem. On instances whose edge weights have been chosen at random, the Blob-coded GA maintains its time advantage, but its results are inferior to those of the edge-set-coded GA, and both GAs are hard pressed to keep up with a simple stochastic hill-climber on all the test instances.