Complexity of spanning tree problems with leaf-dependent objectives

The purpose of this work was to assess the computational complexity of the design of tree networks when the objective function depends on the set of leaves of the tree (i.e., vertices of degree 1, also called “pendant” vertices). To the authors’ knowledge, some of these problems have not been addressed in the literature, although applications come immediately to mind: In fact, it is not rare that certain forms of delivery and/or pickup services may be done at no extra cost as far as intermediate vertices along a path are concerned, thus making the relevant cost contribution depend only on the length of the paths to or between leaves. Further, in public transport analysis, the level of service or users’ satisfaction can be clearly measured by a function of the distances from or between the leaves of a transport tree. By choosing such a type of objectives, the decision maker is more concerned by “equity” since the leaves correspond to the farthest and, thus, most disadvantaged users. Finally, for highly time constrained problems, like newspaper distribution, all customers must be reached within a short time, so the only important vertices in the network correspond to the leaves of the delivery routes. All problems that we considwed are in NPO, the class of nondeterministic polynomial optimization problems (for a precise definition of this class, see, e.g., Crescenzi and Panconesi [ 4 ] ) and therefore are identified by a 4tuple ( I , sol, f , goal), where I is the set of instances of the problem considered; for any instance x, sol(x) is the set of feasible solutions of x ; f ( x, y ) is the objective function evaluating the feasible solution y E so/(x); and goal E { max, min} tells if we are interested into a maximization or a minimization problem. The corresponding decision problem is obtained by letting goal E { 2 k , 512 } , where k is a given constant. Both optimization and decision versions of the same problem are used in what follows. For all the problems considered here, an instance x E I is given by a connected graph G = ( V , E ) with vertex set I.’ = { ‘u,, . . . , on} and edge set E of cardinality m; moreover, sol(x) is the set of spanning trees T of G, i.e., subgraphs having the same set of vertices as G , a set of n 1 edges of E , and no cycles. We denote the set of leaves