Generalized roof duality and bisubmodular functions

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 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 = where {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 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.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Vladimir Kolmogorov
• Keywords: CORR 2010, Education, Pseudo-boolean Functions, Roof Duality, Roof Duality Relaxation