Generalized roof duality and bisubmodular functions

Consider a convex relaxation (f) over cap of a pseudo-Boolean function f. We say that the relaxation is totally half-integral if (f) over cap (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) = xj, x(i) = 1 - x(j), and x(i) = gamma where gamma epsilon {0, 1, 1/2} 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) over cap 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. (c) 2011 Elsevier B.V. All rights reserved.