Fulla, Uppman, and Živný [ACM ToCT'18] established a dichotomy theorem for Boolean surjective general-valued constraint satisfaction problems (VCSPs), i.e., VCSPs on two-element domains in which both labels have to be used in a solution. This result, in addition to identifying the complexity frontier, features the discovery of a new non-trivial tractable case (called EDS) that does not appear in the non-surjective setting. In this work, we go beyond Boolean domains. As our main result, we introduce a generalisation of EDS to arbitrary finite domains called SEDS (similar to EDS) and establish a conditional complexity classification of SEDS VCSPs based on a reduction to smaller domains. This gives a complete classification of SEDS VCSPs on three-element domains. The basis of our tractability result is a natural generalisation of the Min-Cut problem, in which only solutions of certain size (given by a lower and upper bound) are permitted. We show that all near-optimal solutions to this problem can be enumerated in polynomial time, which might be of independent interest.
@InProceedings{matl_et_al:LIPIcs.STACS.2019.52, author = {Matl, Gregor and \v{Z}ivn\'{y}, Stanislav}, title = {{Beyond Boolean Surjective VCSPs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {52:1--52:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.52}, URN = {urn:nbn:de:0030-drops-102911}, doi = {10.4230/LIPIcs.STACS.2019.52}, annote = {Keywords: constraint satisfaction problems, valued constraint satisfaction, surjective constraint satisfaction, graph cuts} }
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