On the Complexity of the One-terminal Network Design Problem

We will consider one-terminal network design problems of the sort of Section A2.1 in Garey and Johnson's book [2]. We will explore the borderline between easy and hard problems of this kind (from the computational complexity point of view). A related problem is the capacitated spanning tree problem (denoted ND5 in [2]). Another related paper is [3] where a linear programming line of attack is adopted. The reader is referred to Section A2.1 in [2] for additional references. The general problem is to select a subset E' c E of edges from a graph G = (V, E ) so that the total cost associated with the subgraph G'= ( V , E') is minimized. The total cost consists of the cost of constructing the roads corresponding to E ' plus the expected cost of traveling within the subnetwork G'. The latter may be discounted over time. We will show that simple subcases of this problem are NP-hard. Thus, the restrictive assumptions that we list below strengthen in fact the NP-hardness results. First, we note that the multi-terminal problem is NP-hard even if all the demands are equal; this follows from the fact that the problem of finding the shortest total path length spanning tree is NP-hard [2, p. 2061. We consider only one-terminal problems, i.e. the demand for traveling is between each vertex v and one distinguished vertex t (the terminal). We assume all demands are equal, i.e., for example, at each vertex v there lives one user who has to commute daily between v and t . Two extreme cases are easy. First, if the cost of commuting is zero, i.e. only construction cost should be minimized, then our problem is that of the minimum spanning tree. Second, if the cost of construction is zero then the total length of paths from all v's to t has to be minimized and this is achieved by the tree of shortest routes from t to all v's. This is because each user travels the shortest distance possible whatsoever. This tree is found very efficiently by Dijstra's shortest-path algorithm (see [I]). If demands are not all equal then minimizing the cost of construction alone if NP-hard even when all demands are either zero or one, since this is in fact the Steiner tree problem in graphs (ND12). The one-terminal problem of minimizing the total weighted path length is solved, of course, by the tree of shortest routes from t to all 0's. We now turn to the case where the two different costs are positive. We further assume that the costs are proportional to the length d(l, j ) of the edge (i, j) . We first claim that the optimal solution can be assumed, without loss of generality, to be a tree. This is argued as follows. Let x,, denote the number of users who will travel, on their way to t , along the edge (i, j ) in the direction from i to j. Since exactly one user travels from I to t (not necessarily along the edge (i, t)), B,(x,, x,,) = 1. Let a denote the cost of constructing a unit-length of road and let b denote the unit travel cost ( a , b > 0). Thus, the total cost associated with the edge ( i , j ) is zero if x,, = 0 and equals ( a + b x,,) d(i, j ) if x,, > 0. We have to minimize the sum of all edge-costs subject to Z,(x,, x,,) = 1, x,, 2 0.