In this work, we first study a natural generalisation of the Min-Cut problem, where a graph is augmented by a superadditive set function defined on its vertex subsets. The goal is to select a vertex subset such that the weight of the induced cut plus the set function value are minimised. In addition, a lower and upper bound is imposed on the solution size. We present a polynomial-time algorithm for enumerating all near-optimal solutions of this Bounded Generalised Min-Cut problem. Second, we apply this novel algorithm to surjective general-valued constraint satisfaction problems (VCSPs), i.e., VCSPs in which each label has to be used at least once. On the Boolean domain, Fulla, Uppman, and Živný (ACM ToCT’18) have recently established a complete classification of surjective VCSPs based on an unbounded version of the Generalised Min-Cut problem. Their result features the discovery of a new non-trivial tractable case called EDS that does not appear in the non-surjective setting. As our main result, we extend the class EDS to arbitrary finite domains and provide a conditional complexity classification for surjective VCSPs of this type based on a reduction to smaller domains. On three-element domains, this leads to a complete classification of such VCSPs.
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It appears to be an issue that the superadditive set function is evaluated only for the solution set, while the set of remaining vertices may exhibit an excessively large set function value even in an optimal solution. That makes it implausible to think a local criterion for edge contractions could incorporate the superadditive set function in a suitable manner, i.e. somehow preventing the set function value from getting too large.
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We would like to thank the anonymous referees of both the conference  and this full version of the paper. We also thank Costin-Andrei Oncescu for detailed feedback on a previous version of this paper.
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An extended abstract of this work appeared in Proceedings of the 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019) . Stanislav Živný was supported by a Royal Society University Research Fellowship. The work was done while Gregor Matl was at the University of Oxford. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 714532). The paper reflects only the authors’ views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
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Matl, G., Živný, S. Using a Min-Cut Generalisation to Go Beyond Boolean Surjective VCSPs. Algorithmica (2020). https://doi.org/10.1007/s00453-020-00735-1
- Constraint satisfaction problems
- Valued constraint satisfaction problems
- Surjective VCSPs