On the Containment Problem for Linear Sets

It is well known that the containment problem (as well as the equivalence problem) for semilinear sets is log-complete in Πp2. In this paper, we show that already the containment problem for linear sets is log-hard (and therefore also log-complete) in Πp2. This holds even when we use a unary encoding for the numerical input parameters or when we restrict the problem to 1-dimensional linear sets (i.e, linear sets in N0). However, combining both restrictions, the problem becomes solvable in polynomial time.