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# Testing the Complexity of a Valued CSP Language

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LIPIcs.ICALP.2019.77.pdf
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## Cite As

Vladimir Kolmogorov. Testing the Complexity of a Valued CSP Language. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 77:1-77:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.77

## Abstract

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 Gamma of cost functions, called a language. Recent breakthrough results have established a complete complexity classification of such classes with respect to language Gamma: if all cost functions in Gamma satisfy a certain algebraic condition then all Gamma-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language Gamma 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 Gamma can be tested in O(sqrt[3]{3}^{|D|}* poly(size(Gamma))) time, where D is the domain of Gamma 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 delta<1 there is no O(sqrt[3]{3}^{delta|D|}) algorithm, assuming that SETH holds.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• Valued Constraint Satisfaction Problems
• Exponential time algorithms
• Exponential Time Hypothesis

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