The threshold order of a Boolean function
Discrete Applied Mathematics
Discrete Mathematics
Equational characterizations of Boolean function classes
Discrete Mathematics
Equivalent Representations of Set Functions
Mathematics of Operations Research
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
A combinatorial strongly polynomial algorithm for minimizing submodular functions
Journal of the ACM (JACM)
Submodularity, Supermodularity, and Higher-Order Monotonicities of Pseudo-Boolean Functions
Mathematics of Operations Research
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We examine classes of real-valued functions of 0-1 variables closed under algebraic operations as well as topological convergence, and having a certain local characteristic (requiring that any function not in the class should have a k-variable minor not belonging to this class). It is shown that for k=2, the only 4 maximal classes with these properties are those of submodular, supermodular, monotone increasing and monotone decreasing functions. All the 13 locally defined closed classes are determined and shown to be intersections of the 4 maximal ones. All maximal classes for k=3 are determined and characterized by the sign of higher order derivatives of the functions in the class.