@article{3257,
abstract = {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 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=γ where γ∈{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 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. 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.},
author = {Kolmogorov, Vladimir},
journal = {Discrete Applied Mathematics},
number = {4-5},
pages = {416 -- 426},
publisher = {Elsevier},
title = {{Generalized roof duality and bisubmodular functions}},
doi = {10.1016/j.dam.2011.10.026},
volume = {160},
year = {2012},
}
@misc{5396,
abstract = {We consider the problem of inference in agraphical model with binary variables. While in theory it is arguably preferable to compute marginal probabilities, in practice researchers often use MAP inference due to the availability of efficient discrete optimization algorithms. We bridge the gap between the two approaches by introducing the Discrete Marginals technique in which approximate marginals are obtained by minimizing an objective function with unary and pair-wise terms over a discretized domain. This allows the use of techniques originally devel-oped for MAP-MRF inference and learning. We explore two ways to set up the objective function - by discretizing the Bethe free energy and by learning it from training data. Experimental results show that for certain types of graphs a learned function can out-perform the Bethe approximation. We also establish a link between the Bethe free energy and submodular functions.},
author = {Korc, Filip and Kolmogorov, Vladimir and Lampert, Christoph},
issn = {2664-1690},
pages = {13},
publisher = {IST Austria},
title = {{Approximating marginals using discrete energy minimization}},
doi = {10.15479/AT:IST-2012-0003},
year = {2012},
}