Submodularity, Supermodularity, and Higher-Order Monotonicities of Pseudo-Boolean Functions

Authors:
Stephan Foldes;Peter L. Hammer
Affiliations:
-;-
Venue:
Mathematics of Operations Research
Year:
2005

Citing 0
Cited 3

Functional equations, constraints, definability of function classes, and functions of Boolean variables

Acta Cybernetica
Algebraic and topological closure conditions for classes of pseudo-Boolean functions

Discrete Applied Mathematics
Locally monotone Boolean and pseudo-Boolean functions

Discrete Applied Mathematics

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Abstract

Classes of set functions defined by the positivity or negativity of the higher-order derivatives of their pseudo-Boolean polynomial representations generalize those of monotone, supermodular, and submodular functions. In this paper, these classes are characterized by functional inequalities and are shown to be closed both under algebraic closure conditions and a local closure criterion. It is shown that for everym = 1, in addition to the class of all set functions, there are only three other classes satisfying these algebraic and local closure conditions: those having positive, respectively negative,mth-order derivatives, and those having a polynomial representation of degree less thanm.