ON COMPLEXITY OF SUBSET INTERCONNECTION DESIGNS

Given a set X and subsets X(1),..,X(m), we consider the problem of finding a graph G with vertex set X and the minimum number of edges such that for i = 1,..., m, the subgraph G(i) induced by X(i) is connected. Suppose that for any alpha points x(1),...,x(alpha), is an element of X, there are at most beta X(i)'s containing the set (x(1),..., x(alpha)). In the paper, we show that the problem is polynomial-time solvable for (alpha less than or equal to 2, beta less than or equal to 2) and is NP-hard for (a greater than or equal to 3, beta = 1), (alpha = 1, beta greater than or equal to 6), and (alpha greater than or equal to 2, beta greater than or equal to 3).