Before revision:  In particular (  in contrast to the cardinality-constrained case), there cannot be an instance where (i) it is impossible to achieve a factor better than 1/2 regardless of the number of rounds, and (ii) it takes r rounds to achieve a factor of 1/2-O(1/r).
After revision:  In particular ( in contrast to the cardinality-constrained case), there cannot be an instance where (i) it is impossible to achieve an approximation factor better than 1/2 regardless of the number of rounds, and (ii) it takes r rounds to achieve a factor of 1/2-O(1/r).