Testing the complexity of a valued CSP language LIPIcs Kolmogorov, Vladimir ddc:000 A Valued Constraint Satisfaction Problem (VCSP) provides a common framework that can express a wide range of discrete optimization problems. A VCSP instance is given by a finite set of variables, a finite domain of labels, and an objective function to be minimized. This function is represented as a sum of terms where each term depends on a subset of the variables. To obtain different classes of optimization problems, one can restrict all terms to come from a fixed set Γ of cost functions, called a language. Recent breakthrough results have established a complete complexity classification of such classes with respect to language Γ: if all cost functions in Γ satisfy a certain algebraic condition then all Γ-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language Γ is known to be NP-hard. We thus study exponential algorithms for this meta-problem. We show that the tractability condition of a finite-valued language Γ can be tested in O(3‾√3|D|⋅poly(size(Γ))) time, where D is the domain of Γ and poly(⋅) is some fixed polynomial. We also obtain a matching lower bound under the Strong Exponential Time Hypothesis (SETH). More precisely, we prove that for any constant δ<1 there is no O(3‾√3δ|D|) algorithm, assuming that SETH holds. Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2019 info:eu-repo/semantics/conferenceObject doc-type:conferenceObject text http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/6725 https://research-explorer.ista.ac.at/download/6725/6738 Kolmogorov V. Testing the complexity of a valued CSP language. In: <i>46th International Colloquium on Automata, Languages and Programming</i>. Vol 132. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2019:77:1-77:12. doi:<a href="https://doi.org/10.4230/LIPICS.ICALP.2019.77">10.4230/LIPICS.ICALP.2019.77</a> eng info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPICS.ICALP.2019.77 info:eu-repo/semantics/altIdentifier/issn/1868-8969 info:eu-repo/semantics/altIdentifier/isbn/978-3-95977-109-2 info:eu-repo/semantics/altIdentifier/arxiv/1803.02289 info:eu-repo/semantics/openAccess