Greedy heuristics and an evolutionary algorithm for the bounded-diameter minimum spanning tree problem

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 lowest weight in which no path between two vertices contains more than D edges. This problem is NP-hard for 4 < D < n - 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 lowest-weight edges that it can, but such edges do not in general connect the interior nodes of low-weight bounded-diameter trees. A new randomized greedy heuristic builds a bounded-diameter spanning tree from its center vertex or vertices. It chooses each next vertex at random but attaches the vertex with the lowest-weight 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 offspring trees. These operators are efficient, so the algorithm scales well to larger problem instances. On 25 Euclidean instances of up to 1000 vertices, the EA improved substantially on solutions found by the randomized greedy heuristic.