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    The complexity of solution-free sets of integers for general linear equations

    Edwards, K. and Noble, Steven (2019) The complexity of solution-free sets of integers for general linear equations. Discrete Applied Mathematics 270 , pp. 115-133. ISSN 0166-218X.

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    Given a linear equationL, a setAof integers isL-free ifAdoes not contain anynon-trivial solutions toL. Meeks and Treglown [6] showed that for certain kindsof linear equations, it isNP-complete to decide if a given set of integers containsa solution-free subset of a given size. Also, for equations involving three variables,they showed that the problem of determining the size of the largest solution-freesubset isAPX-hard, and that for two such equations (representing sum-free andprogression-free sets), the problem of deciding if there is a solution-free subset withat least a specified proportion of the elements is alsoNP-complete.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 solution-free subsets ofa given set, and show that this problem is #P-complete for any linear equation inat least three variables.


    Item Type: Article
    Keyword(s) / Subject(s): solution-free 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


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