Generalized roof duality and bisubmodular functions

https://doi.org/10.1016/j.dam.2011.10.026Get rights and content
Under an Elsevier user license
open archive

Abstract

Consider a convex relaxation fˆ of a pseudo-Boolean function f. We say that the relaxation is totally half-integral if fˆ(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 xi=xj, xi=1xj, and xi=γ where γ{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 fˆ 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.

Keywords

Pseudo-boolean optimization
Roof duality
Bisubmodularity

Cited by (0)

A preliminary version of this paper appeared in the proceedings of Neural Information Processing Systems conference (NIPS), 2010.