Matroids and Subset Interconnection Design

Abstract

A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows. (1) Given a set X and subsets $X_i ,Y_i $ of $X,i = 1, \cdots ,n$, satisfying $X_i \cap Y_i = \O $, find a graph G with vertex set X and the minimum number of edges such that for any i, the subgraph induced by $X\backslash Y_i $ has a connected component containing $X_i $. Two other problems related to this one are the following ones. (2) Given a set X and subsets $X_1 ,X_2 , \cdots ,X_n $ such that $X = \cup _{i = 1}^n X_i $, find a graph G with vertex set X and the minimum number of edges such that for any i the subgraph $G_i $ induced by $X_i $ in G is connected. (3) Given a set X and subsets $X_1 ,X_2 , \cdots ,X_n $ such that $X = \cup _{i = 1}^n X_i $, find a graph G with vertex set X, find a graph G with vertex set X and the minimum number of edges such that for any subset I of $\{ 1, \cdots ,n \}$, the subgraph induced by $ \cap _{i \in I} X_i $ is connected. This paper shows that (3) is polynomial-time solvable while (1) and (2) are NP-complete. Also, some heuristics for (1) and (2) are given. The solution of (3) is an interesting application of matroid theory.