We study the problem of anonymizing tables containing personal information before releasing them for public use. One of the formulations considered in this context is the $k$-anonymization problem: given a table, suppress a minimum number of cells so that in the transformed table, each row is identical to atleast $k-1$ other rows. The problem is known to be NP-hard and MAXSNP-hard; but in the known reductions, the number of columns in the constructed tables is arbitrarily large. However, in practical settings the number of columns is much smaller. So, we study the complexity of the practical setting in which the number of columns $m$ is small. We show that the problem is NP-hard, even when the number of columns $m$ is a constant ($m=3$). We also prove MAXSNP-hardness for this restricted version and derive that the problem cannot be approximated within a factor of (6238/6237). Our reduction uses alphabets $Σ$ of arbitrarily large size. A natural question is whether the problem remains NP-hard when both $m$ and $|Σ|$ are small. We prove that the $k$-anonymization problem is in $P$ when both $m$ and $|Σ|$ are constants.