Disjunctive analogues of submodular and supermodular pseudo-Boolean functions

AbstractWe consider classes of real-valued functions of Boolean variables defined by disjunctive analogues of the submodular and supermodular functional inequalities, obtained by replacing in these inequalities addition by disjunction (max operator). The disjunctive analogues of submodular and supermodular functions are completely characterized by the syntax of their disjunctive normal forms. Classes of functions possessing combinations of these properties are also examined. A disjunctive representation theory based on one of these combination classes exhibits syntactic and algorithmic analogies with classical DNF theory

Equational characterization of Boolean function classes

Cataloged from PDF version of article.Several noteworthy classes of Boolean functions can be characterized by algebraic identities (e.g. the class of positive functions consists of all functions f satisfying the identity f(x) V f(y) V f(x V y) = f(x V y)). We give algebraic identities for several of the most frequently analyzed classes of Boolean functions (including Horn, quadratic, supermodular, and submodular functions) and proceed then to the general question of which classes of Boolean functions can be characterized by algebraic identities. We answer this question for function classes closed under addition of inessential (irrelevant) variables. Nearly all classes of interest have this property. We show that a class with this property has a characterization by algebraic identities if and only if the class is closed under the operation of variable identification. Moreover, a single identity suffices to characterize a class if and only if the number of minimal forbidden identification minors is finite. Finally, we consider characterizations by general first-order sentences, rather than just identities. We show that a class of Boolean functions can be described by an appropriate set of such first-order sentences if and only if it is closed under permutation of variables. © 2000 Elsevier Science B.V. All rights reserved

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum