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Definable inapproximability: New challenges for duplicator

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Atserias, A 

Abstract

jats:titleAbstract</jats:title>jats:pWe 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 $\textrm{P} = \textrm{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.</jats:p>

Description

Keywords

Descriptive Complexity, Hardness of Approximation, MAX SAT, Vertex Cover, Fixed-point Logic with Counting

Journal Title

Journal of Logic and Computation

Conference Name

Journal ISSN

0955-792X
1465-363X

Volume Title

29

Publisher

Oxford University Press (OUP)

Rights

All rights reserved
Sponsorship
Engineering and Physical Sciences Research Council (EP/S03238X/1)