Minimum-weight two-connected spanning networks

Abstract

We consider the problem of constructing a minimum-weight, two-connected network spanning all the points in a setV. We assume a symmetric, nonnegative distance functiond(·) defined onV × V which satisfies the triangle inequality. We obtain a structural characterization of optimal solutions. Specifically, there exists an optimal two-connected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since we also show thatany two-connected graph satisfying these conditions is theunique optimal solution for a particular choice of ‘canonical’ distances satisfying the triangle inequality. We use these properties to show that the weight of an optimal traveling salesman cycle is at most 4/3 times the weight of an optimal two-connected solution; examples are provided which approach this bound arbitrarily closely. In addition, we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points.