On regularity of Max-CSPs and Min-CSPs

•Regular Max-CSPs have the same (up to o(1) error) approximability as Max-CSPs.•Regular Min-CSPs have the same (up to o(1) error) approximability as Min-CSPs.•Weights do not matter (up to o(1) error) for approximability of Max/Min-CSPs.•Constant degree regular instances essentially capture approximability of Max-CSPs. We study the approximability of regular constraint satisfaction problems, i.e., CSPs where each variable in an instance has the same number of occurrences. In particular, we show that for any CSP Λ, the existence of an α-approximation algorithm for unweighted regular Max-CSP Λ implies the existence of an (α−o(1))-approximation algorithm for weighted Max-CSP Λ for which the regularity of instances is not imposed. We also give an analogous result for Min-CSPs, and therefore show that up to an arbitrarily small error it is sufficient to conduct the study of the approximability of CSPs only on regular unweighted instances.