Integer and combinatorial optimization
Integer and combinatorial optimization
A new approach to the maximum-flow problem
Journal of the ACM (JACM)
On the supermodular knapsack problem
Mathematical Programming: Series A and B
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
Complexity classifications of boolean constraint satisfaction problems
Complexity classifications of boolean constraint satisfaction problems
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
Submodular function minimization
Mathematical Programming: Series A and B
A Faster Strongly Polynomial Time Algorithm for Submodular Function Minimization
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Valued constraint satisfaction problems: hard and easy problems
IJCAI'95 Proceedings of the 14th international joint conference on Artificial intelligence - Volume 1
Constraints, consistency and closure
Artificial Intelligence
The complexity of soft constraint satisfaction
Artificial Intelligence
Supermodular functions and the complexity of MAX CSP
Discrete Applied Mathematics - Special issue: Boolean and pseudo-boolean funtions
An algebraic characterisation of complexity for valued constraint
CP'06 Proceedings of the 12th international conference on Principles and Practice of Constraint Programming
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Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial time. However, the general algorithm is of order O(n6) and hence rather impractical. In this paper, by using results from the theory of pseudo-Boolean optimisation, we identify several broad classes of submodular constraints over a Boolean domain which are expressible using binary submodular constraints, and hence can be minimised in cubic time. We also discuss the question of whether all submodular constraints of bounded arity over a Boolean domain are expressible using only binary submodular constraints, and can therefore be minimised efficiently.