Definable Inapproximability: New Challenges for Duplicator

Authors Albert Atserias , Anuj Dawar

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Albert Atserias
  • Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain
Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK

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Albert Atserias and Anuj Dawar. Definable Inapproximability: New Challenges for Duplicator. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Finite Model Theory
  • Descriptive Compleixty
  • Hardness of Approximation
  • Vertex Cover
  • Fixed-point logic with counting


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