Node Coincidence in Metric Minimum Weighted Length Graph Embeddings

The minisum multifacility location problem is viewed as finding an embedding of a graph in a metric space under additional constraints such as a number of fixed vertex locations, minimising the sum of weighted lengths of all edges. We show that certain nontrivial minimal cuts in the graph are sets of nodes that will necessarily coincide at any or at some optimal solution, irrespective of the fixed locations and the metric. This new property strongly generalises all coincidence conditions known in literature. In fact we show that it is best possible for coincidence with a fixed vertex at any position in arbitrary metric spaces. For coincidence among free vertices a different property of graph symmetry is also sufficient, and we conjecture its best possibility in conjunction with the minimal cut condition. All such instance-independent coincidences (both in at least one and in all optimal solutions) may be determined efficiently.