Definable Inapproximability: New Challenges for Duplicator
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.
Descriptive Compleixty
Hardness of Approximation
MAX SAT
Vertex Cover
Fixed-point logic with counting
Theory of computation~Complexity theory and logic
Theory of computation~Finite Model Theory
7:1-7:21
Regular Paper
https://arxiv.org/abs/1806.11307
Albert
Atserias
Albert Atserias
Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain
https://orcid.org/0000-0002-3732-1989
Partially funded by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement ERC-2014-CoG 648276 (AUTAR) and MICCIN grant TIN2016-76573-C2-1P (TASSAT3).
Anuj
Dawar
Anuj Dawar
Department of Computer Science and Technology, University of Cambridge, UK
https://orcid.org/0000-0003-4014-8248
Supported in part by a Fellowship of the Alan Turing Institute.
10.4230/LIPIcs.CSL.2018.7
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Albert Atserias and Anuj Dawar
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