The complexity of solution-free sets of integers

TL;DR

This paper extends the complexity results of solution-free sets of integers to all linear equations, showing that key decision and counting problems remain computationally hard even under bounded conditions.

Contribution

It generalizes previous hardness results to all linear equations and bounded sets, and addresses the complexity of counting solution-free subsets.

Findings

Deciding the existence of large solution-free subsets is NP-complete.

Counting the number of solution-free subsets is #P-complete for equations with three or more variables.

Hardness results hold even for sets with polynomially bounded positive integers.

Abstract

Given a linear equation L, a set A of integers is L-free if A does not contain any non-trivial solutions to L. Meeks and Treglown showed that for certain kinds of linear equations, it is NP-complete to decide if a given set of integers contains a 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-free subset is APX-hard, and that for two such equations (representing sum-free and progression-free sets), the problem of deciding if there is a solution-free subset with at least a specified proportion of the elements is also NP-complete. We answer a number of questions posed by Meeks and Treglown, by extending the results above to all linear equations, and showing that the problems remain hard for 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 coefficients do not all have the same sign. We also consider the problem of counting the number of solution-free subsets of a given set, and show that this problem is #P-complete for any linear equation in at least three variables.