Generalized roof duality and bisubmodular functions

Consider a convex relaxation $\hat f$ of a pseudo-boolean function $f$. We say that the relaxation is {\em totally half-integral} if $\hat f(x)$ 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 $x_i=\gamma$ where \gamma\in\{0, 1, 1/2} is a constant. A well-known example is the {\em 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 $\hat f$ by establishing a one-to-one correspondence with {\em 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.