Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains
Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. It is desirable to classify the computational complexity of VCSPs depending on a fixed set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified in this sense. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear homogeneous cost functions. We remark that in this paper the infinite domain will always be the set of rational numbers. We show that such VCSPs can be solved in polynomial time when the cost functions are additionally submodular, and that this is indeed a maximally tractable class: adding any cost function that is not submodular leads to an NP-hard VCSP.
Valued constraint satisfaction problems
Piecewise linear functions
Submodular functions
Semilinear
Constraint satisfaction
Optimisation
Model Theory
Mathematics of computing~Mathematical optimization
Theory of computation~Complexity theory and logic
12:1-12:22
Regular Paper
Manuel
Bodirsky
Manuel Bodirsky
Institut für Algebra, Technische Universität Dresden, Germany
The first and second author have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 681988, CSP-Infinity). The first author has also received funding from the DFG (Project number 622397)
Marcello
Mamino
Marcello Mamino
Dipartimento di Matematica, Università di Pisa, Italy
Caterina
Viola
Caterina Viola
Institut für Algebra, Technische Universität Dresden, Germany
This author is supported by DFG Graduiertenkolleg 1763 (QuantLA).
10.4230/LIPIcs.CSL.2018.12
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Manuel Bodirsky, and Marcello Mamino, and Caterina Viola
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