Integer and combinatorial optimization
Integer and combinatorial optimization
On the supermodular knapsack problem
Mathematical Programming: Series A and B
Recognition problems for special classes of polynomials in 0-1 variables
Mathematical Programming: Series A and B
Perspectives of Monge properties in optimization
Discrete Applied Mathematics
Constraints, consistency and closure
Artificial Intelligence
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
A combinatorial strongly polynomial algorithm for minimizing submodular functions
Journal of the ACM (JACM)
A fully combinatorial algorithm for submodular function minimization
Journal of Combinatorial Theory Series B
Discrete Applied Mathematics
What Energy Functions Can Be Minimizedvia Graph Cuts?
IEEE Transactions on Pattern Analysis and Machine Intelligence
Classifying the Complexity of Constraints Using Finite Algebras
SIAM Journal on Computing
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
Supermodular functions and the complexity of MAX CSP
Discrete Applied Mathematics - Special issue: Boolean and pseudo-boolean funtions
The Approximability of Three-valued MAX CSP
SIAM Journal on Computing
The complexity of soft constraint satisfaction
Artificial Intelligence
Level of repair analysis and minimum cost homomorphisms of graphs
Discrete Applied Mathematics
Handbook of Constraint Programming (Foundations of Artificial Intelligence)
Handbook of Constraint Programming (Foundations of Artificial Intelligence)
A Linear Programming Approach to Max-Sum Problem: A Review
IEEE Transactions on Pattern Analysis and Machine Intelligence
Submodular function minimization
Mathematical Programming: Series A and B
Maximizing Non-Monotone Submodular Functions
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
SIAM Journal on Discrete Mathematics
The approximability of MAX CSP with fixed-value constraints
Journal of the ACM (JACM)
A faster strongly polynomial time algorithm for submodular function minimization
Mathematical Programming: Series A and B
A simple combinatorial algorithm for submodular function minimization
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Graphical Models, Exponential Families, and Variational Inference
Foundations and Trends® in Machine Learning
Robust Higher Order Potentials for Enforcing Label Consistency
International Journal of Computer Vision
A maximal tractable class of soft constraints
Journal of Artificial Intelligence Research
Valued constraint satisfaction problems: hard and easy problems
IJCAI'95 Proceedings of the 14th international joint conference on Artificial intelligence - Volume 1
An algebraic characterisation of complexity for valued constraint
CP'06 Proceedings of the 12th international conference on Principles and Practice of Constraint Programming
Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
IEEE Transactions on Pattern Analysis and Machine Intelligence
An algebraic theory of complexity for valued constraints: establishing a Galois connection
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Minimizing a sum of submodular functions
Discrete Applied Mathematics
Discrete Applied Mathematics
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
The expressibility of functions on the boolean domain, with applications to counting CSPs
Journal of the ACM (JACM)
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We investigate whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This question has been considered in several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively. Our results have several corollaries. First, we characterise precisely which submodular polynomials of arity 4 can be expressed by binary submodular polynomials. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish, for the first time, that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.