On Regularity of Max-CSPs and Min-CSPs

We study 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 $\Lambda$, existence of an $\alpha$ approximation algorithm for unweighted regular Max-CSP $\Lambda$ implies existence of an $\alpha-o(1)$ approximation algorithm for weighted Max-CSP $\Lambda$ in which regularity of the instances is not imposed. We also give an analogous result for Min-CSPs, and therefore show that up to arbitrarily small error it is sufficient to conduct the study of approximability of CSPs only on regular unweighted instances.