Modular Decomposition of Boolean Functions

Modular decomposition is a thoroughly investigated topic in many areas such as switching theory, reliability theory, game theory and graph theory. Most appli- cations can be formulated in the framework of Boolean functions. In this paper we give a uni_ed treatment of modular decomposition of Boolean functions based on the idea of generalized Shannon decomposition. Furthermore, we discuss some new results on the complexity of modular decomposition. We propose an O(mn)- algorithm for the recognition of a modular set of a monotone Boolean function f with m prime implicants and n variables. Using this result we show that the the computation of the modular closure of a set can be done in time O(mn2). On the other hand, we prove that the recognition problem for general Boolean functions is coNP-complete

Bimatroidal independence systems

peer reviewedAn independence system Σ=(X, F) is called bimatroidal if there exist two matroids M=(X,FM) and N=(X,FN) such that F=FM∪FN. When this is the case, {M,N} is called a bimatroidal decomposition of Σ. This paper initiates the study of bimatroidal systems. Given the collection of circuits of an arbitrary independence system Σ (or, equivalently, the constraints of a set-covering problem), we address the following question: does Σ admit a bimatroidal decomposition {M,N} and, if so, how can we actually produce the circuits of M andN? We derive a number of results concerning this problem, and we present a polynomial time algorithm to solve it when every two circuits of Σ have at most one common element. We also propose different polynomial time algorithms for set covering problems defined on the circuit-set of bimatroidal systems