The complexity of solutionfree sets of integers for general linear equations
Edwards, K. and Noble, Steven (2019) The complexity of solutionfree sets of integers for general linear equations. Discrete Applied Mathematics 270 , pp. 115133. ISSN 0166218X.

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Abstract
Given a linear equationL, a setAof integers isLfree ifAdoes not contain anynontrivial solutions toL. Meeks and Treglown [6] showed that for certain kindsof linear equations, it isNPcomplete to decide if a given set of integers containsa solutionfree subset of a given size. Also, for equations involving three variables,they showed that the problem of determining the size of the largest solutionfreesubset isAPXhard, and that for two such equations (representing sumfree andprogressionfree sets), the problem of deciding if there is a solutionfree subset withat least a specified proportion of the elements is alsoNPcomplete.We answer a number of questions posed by Meeks and Treglown, by extendingthe results above to all linear equations, and showing that the problems remain hardfor sets of integers whose elements are polynomially bounded in the size of the set.For most of these results, the integers can all be positive as long as the coefficientsdo not all have the same sign.We also consider the problem of counting the number of solutionfree subsets ofa given set, and show that this problem is #Pcomplete for any linear equation inat least three variables.
Metadata
Item Type:  Article 

Keyword(s) / Subject(s):  solutionfree set, computational complexity 
School:  Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences 
Depositing User:  Steven Noble 
Date Deposited:  30 Jul 2019 15:07 
Last Modified:  09 Aug 2023 12:46 
URI:  https://eprints.bbk.ac.uk/id/eprint/28258 
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