Generalized roof duality and bisubmodular functions

Consider a convex relaxation f̂ of a pseudo-boolean function f . We say that the relaxation is totally half-integral if f̂(x) is a polyhedral function with halfintegral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form xi = xj , xi = 1 − xj , 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 f̂ 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.