Roof duality for polynomial 0-1 optimization
Mathematical Programming: Series A and B
Recognition problems for special classes of polynomials in 0-1 variables
Mathematical Programming: Series A and B
Persistency in quadratic 0–1 optimization
Mathematical Programming: Series A and B
Fast Approximate Energy Minimization via Graph Cuts
IEEE Transactions on Pattern Analysis and Machine Intelligence
Discrete Applied Mathematics
Energy Minimization via Graph Cuts: Settling What is Possible
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 2 - Volume 02
Bisubmodular Function Minimization
SIAM Journal on Discrete Mathematics
Minimizing Nonsubmodular Functions with Graph Cuts-A Review
IEEE Transactions on Pattern Analysis and Machine Intelligence
P³ & Beyond: Move Making Algorithms for Solving Higher Order Functions
IEEE Transactions on Pattern Analysis and Machine Intelligence
Note: The expressive power of binary submodular functions
Discrete Applied Mathematics
Global Stereo Reconstruction under Second-Order Smoothness Priors
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fusion Moves for Markov Random Field Optimization
IEEE Transactions on Pattern Analysis and Machine Intelligence
Graph cut based inference with co-occurrence statistics
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part V
Transformation of General Binary MRF Minimization to the First-Order Case
IEEE Transactions on Pattern Analysis and Machine Intelligence
Curvature regularization for curves and surfaces in a global optimization framework
EMMCVPR'11 Proceedings of the 8th international conference on Energy minimization methods in computer vision and pattern recognition
Global Interactions in Random Field Models: A Potential Function Ensuring Connectedness
SIAM Journal on Imaging Sciences
Generalized roof duality and bisubmodular functions
Discrete Applied Mathematics
Statistical priors for efficient combinatorial optimization via graph cuts
ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part III
Efficient belief propagation with learned higher-order markov random fields
ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part II
Inference for order reduction in Markov random fields
CVPR '11 Proceedings of the 2011 IEEE Conference on Computer Vision and Pattern Recognition
Generalized roof duality for pseudo-boolean optimization
ICCV '11 Proceedings of the 2011 International Conference on Computer Vision
A graph cut algorithm for higher-order Markov Random Fields
ICCV '11 Proceedings of the 2011 International Conference on Computer Vision
Generalized roof duality for multi-label optimization: optimal lower bounds and persistency
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part VI
Tighter relaxations for higher-order models based on generalized roof duality
ECCV'12 Proceedings of the 12th international conference on Computer Vision - Volume Part III
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The roof dual bound for quadratic unconstrained binary optimization is the basis for several methods for efficiently computing the solution to many hard combinatorial problems. It works by constructing the tightest possible lower-bounding submodular function, and instead of minimizing the original objective function, the relaxation is minimized. However, for higher-order problems the technique has been less successful. A standard technique is to first reduce the problem into a quadratic one by introducing auxiliary variables and then apply the quadratic roof dual bound, but this may lead to loose bounds. We generalize the roof duality technique to higher-order optimization problems. Similarly to the quadratic case, optimal relaxations are defined to be the ones that give the maximum lower bound. We show how submodular relaxations can efficiently be constructed in order to compute the generalized roof dual bound for general cubic and quartic pseudo-boolean functions. Further, we prove that important properties such as persistency still hold, which allows us to determine optimal values for some of the variables. From a practical point of view, we experimentally demonstrate that the technique outperforms the state of the art for a wide range of applications, both in terms of lower bounds and in the number of assigned variables.