Generalized roof duality and bisubmodular functions

• Authors: Vladimir Kolmogorov
• Venue: Discrete Applied Mathematics
• Year: 2012

Consider a convex relaxation f@? of a pseudo-Boolean function f. We say that the relaxation is totally half-integral if f@?(x) is a polyhedral function with half-integral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form x"i=x"j, x"i=1-x"j, and x"i=@c where @c@?{0,1,12} is a constant. A well-known example is the roof duality relaxation for quadratic pseudo-Boolean functions f. We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-Boolean functions. Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations f@? by establishing a one-to-one correspondence with bisubmodular functions. Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality. On the conceptual level, our results show that bisubmodular functions provide a natural generalization of the roof duality approach to higher-order terms. This can be viewed as a non-submodular analogue of the fact that submodular functions generalize the s-t minimum cut problem with non-negative weights to higher-order terms.